PL9fwy3NUQKwaIc7KWc8buW344941GStQL/S5DZZsmjnq4.srt

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The second material exam. Question number one. a

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corporation randomly selected or selects 150 

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salespeople and finds that 66% who have never 

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taken self-improvement course would like such a 

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course. So in this case, currently, they select

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150 salespeople and find that 66% would 

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like or who have never taken this course. The firm 

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did a similar study 10 years ago in which 60% of a

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random sample of 160 salespeople wanted a self 

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-improvement course.

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They select a random sample of 160 and tell that 

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60% would like to take this course. So we have

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here two information about previous study and 

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currently. So currently we have this information. 

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The sample size was 150, with a proportion 66% for 

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the people who would like to attend or take this

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course. Mid-Paiwan and Pai Tu represent the true 

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proportion, it means the population proportion, of 

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workers who would like to attend a self 

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-improvement course in the recent study and the 

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past studies in Taiwan. So recent, Paiwan. 

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And Pi 2 is the previous study. This weather, this 

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proportion has changed from the previous study by

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using two approaches. Critical value approach and 

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B value approach. So here we are talking about Pi

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1 equals Pi 2. Since the problem says that The 

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proportion has changed. You don't know the exact 

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direction, either greater than or smaller than. So

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this one should be Y1 does not equal Y2. So step 

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one, you have to state the appropriate null and

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alternative hypothesis.

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Second step, compute the value of the test

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statistic. In this case, your Z statistic should 

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be P1 minus P2 minus Pi 1 minus Pi 2, under the 

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square root of P dash 1 minus P dash times 1 over

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N1 plus 1 over N1. Now, P1 and P2 are given under 

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the null hypothesis Pi 1 minus Pi 2 is 0. So here

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we have to compute P dash, which is the overall 

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B dash equals x1 plus x2 divided by n1 plus n2.

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Now these x's, I mean the number of successes are 

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not given directly in this problem, but we can 

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figure out the values of x1 and x2 by using this 

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information, which is n1 equals 150 and b1 equals 

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66%. Because we know that b1 equals x1 over n1. 

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So, by using this equation, X1 equals N1 times V1. 

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N1 150 times 66 percent, that will give 150 times

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66, so that's 99. So 150 times, it's 99.

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Similarly, X2 equals N2 times V2. N2 is given by 

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160, so 160 times 60 percent, 

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96. So the number of successes are 96 for the

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second, for the previous. Nine nine.

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So B dash equals x1 99

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plus 96 divided by n1 plus n2, 350. And that will

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give the overall proportions divided by 310, 0

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.629.

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So, this is the value of the overall proportion.

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Now, B dash equals 1.629. So, 1 times 1 minus B 

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dash is 1 minus this value times 1 over N1, 1 over

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150 plus 1 over 160. Simple calculation will give

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The value of z, which is in this case 1.093.

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So just plug this information into this equation,

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you will get z value, which is 1.093. He asked to 

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do this problem by using two approaches, critical 

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value and b value. Let's start with the first one,

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b value approach.

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Now your B value or critical value, start with

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critical value. 

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Now since we are taking about a two-sided test, so 

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there are two critical values which are plus or 

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minus Z alpha over. Alpha is given by five

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percent, so in this case

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is equal to plus or minus 1.96.

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Now, does this value, I mean does the value of

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this statistic which is 1.093 fall in the critical 

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region? Now, my critical regions are above 196 or

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below negative 1.96. Now this value actually falls 

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In the non-rejection region, so we don't reject

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the null hypothesis. So my decision, don't reject

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the null hypothesis. That means there is not 

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sufficient evidence to support the alternative

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which states that the proportion has changed from 

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the previous study. So we don't reject the null

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hypothesis. It means there is not sufficient 

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evidence to support the alternative hypothesis.

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That means you cannot say that the proportion has

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changed from the previous study. That by using

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critical value approach. Now what's about p-value? 

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In order to determine the p-value,

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We have to find the probability that the Z

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statistic fall in the rejection regions. So that 

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means Z greater than my values 1093 or 

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Z smaller than negative 1.093. 

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1093 is the same as the left of negative, so they 

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are the same because of symmetry. So just take 1 

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and multiply by 2. 

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Now simple calculation will give the value of 0

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.276 in chapter 6. So go back to chapter 6 to

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figure out how can we calculate the probability of

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Z greater than 1.0938. Now my B value is 0.276, 

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always we reject the null hypothesis if my B value

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is smaller than alpha. Now this value is much much 

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bigger than alpha, so we don't reject the null 

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hypothesis. So since my B value is much greater

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than alpha, that means we don't reject the null 

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hypothesis, so we reach the same conclusion, that

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there is not sufficient evidence to support the 

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alternative. Also, we can perform the test by 

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using confidence interval approach, because here

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we are talking about two-tailed test. Your

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confidence interval is given by

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B1 minus B2 plus 

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or minus Z alpha over 2 times B

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dash 1 minus B dash multiplied by 1 over N1 plus 1 

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over N2. By the way, this one 

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called the margin of error. So z times square root

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of this sequence is called the margin of error, 

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and the square root itself is called the standard 

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error of the point estimate of pi 1 minus pi 2, 

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which is P1 minus P2. So square root of b dash 1 

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minus b dash multiplied by 1 over n1 plus 1 over

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n2 is called the standard error of the estimate of

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pi 1 minus pi 2. So this is standard estimate of

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b1 minus b2. Simply, you will get the confidence 

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interval to be between pi 1 minus the difference 

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between the two proportions, 4 between negative. 0

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.5 and

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0.7. 

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Now this interval actually contains

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The value of 0, that means we don't reject the 

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null hypothesis. So since this interval starts

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from negative, lower bound is negative 0.5, upper 

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bound is 0.17, that means 0 inside this interval,

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I mean the confidence captures the value of 0,

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that means we don't reject the null hypothesis. So

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by using three different approaches, we end with 

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the same decision and conclusion. That is, we 

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don't reject null hypotheses. That's all for

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number one.

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Question number two.

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The excellent drug company claims its aspirin

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tablets will relieve headaches faster than any 

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other aspirin on the market. So they believe that

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Their drug is better than the other drug in the 

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market. To determine whether Excellence claim is

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valid, random samples of size 15 are chosen from 

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aspirins made by Excellence and the sample drug 

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combined. So sample sizes of 15 are chosen from 

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each. So that means N1 equals 15 and N2 also

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equals 15. And aspirin is given to each of the 30

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randomly selected persons suffering from 

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headaches. So the total sample size is 30, because 

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15 from the first company, and the second for the 

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simple company. So they are 30 selected persons

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who are suffering from headaches. So we have 

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information about number of minutes required for

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each to recover from the headache. is recorded,

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the sample results are. So here we have two

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groups, two populations. Company is called 

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excellent company and other one simple company.

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The information we have, the sample means are 8.4

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for the excellent and 8.9 for the simple company.

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With the standard deviations for the sample are 2 

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.05 and 2.14 respectively for excellent and simple

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and as we mentioned the sample sizes are the same

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are equal 15 and 15. Now we are going to test at 

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five percent level of significance test whether to

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determine whether excellence aspirin cure 

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headaches significantly faster than simple

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aspirin. Now faster it means Better. Better it 

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means the time required to relieve headache is

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smaller there. So you have to be careful in this

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case. If we assume that Mu1 is the mean time

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required for excellent aspirin. So Mu1 for

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excellent. 

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So Me1, mean time required for excellence aspirin,

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and Me2, mean time required for simple aspirin. So

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each one, Me1, is smaller than Me3. 

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Since Me1 represents the time required to relieve

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headache by using excellent aspirin and this one

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is faster faster it means it takes less time in

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order to recover from headache so mu1 should be

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smaller than mu2 we are going to use T T is x1 bar

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minus x2 bar minus the difference between the two 

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population proportions divided by

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S squared B times 1 over N1 plus 1 over N2.

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S squared B N1

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minus 1 S1 squared plus N2 minus 1 S2 squared 

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divided by N1 plus N2 minus 1. Now, a simple 

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calculation will give the following results.

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So again, we have this data. Just plug this

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information here to get the value

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Or you maybe use the B-value approach.

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Now, since the alternative is µ1 smaller than µ2,

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So B value is probability of T smaller than

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negative 0

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.653.

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So we are looking for this probability B of Z

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smaller than negative 0.653.

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The table you have gives the area in the upper

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tail. So this is the same as beauty greater than.

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Because the area to the right of 0.653 is the same

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as the area to the left of negative 0.75. Because

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of symmetry. Just look at the tea table. Now,

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smaller than negative, means this area is actually

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the same as the area to the right of the same

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value, but on the other side. So these two areas

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are the same. So it's the same as D of T greater

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than 0.653. If you look at the table for 28

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degrees of freedom,

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That's your 28.

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I am looking for the value of 0.653. The first

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value here is 0.683. The other one is 0.8. It

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means my value is below this one. If you go back

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here,

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so it should be to the left of this value. Now

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here 25, then 20, 20, 15 and so on. So it should

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be greater than 25. So your B value actually is

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greater than 25%. As we mentioned before, T table

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does not give the exact B value. So approximately

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my B value is greater than 25%. This value

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actually is much bigger than 5%. So again, we

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reject, we don't reject the null hypothesis. So

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again, to compute the B value, it's probability of

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T smaller than the value of the statistic, which

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is negative 0.653. The table you have gives the

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area to the right.

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So this probability is the same as B of T greater

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than 0.653. So by using this table, you will get

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approximate value of B, which is greater than 25%.

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Always, as we mentioned, we reject the null

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hypothesis if my B value is smaller than alpha. In

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this case, this value is greater than alpha, so we

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don't reject the null. So we reach the same

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decision as by using the critical value approach.

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Any question? So that's for number two. Question

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number three.

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To test the effectiveness of a business school

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preparation course, eight students took a general

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business test before and after the course. Let X1

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denote before,

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and X2 after.

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And the difference is X2 minus X1.

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The mean of the difference equals 50. And the

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standard deviation of the difference is 65.03. So

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sample statistics are sample mean for the

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difference and sample standard deviation of the

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difference. So these two values are given. Test to

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determine the effectiveness of a business school

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preparation course. So what's your goal? An

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alternative, null equals zero. An alternative

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should

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be greater than zero. Because D is X2 minus X1. So

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effective, it means after is better than before.

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So my score after taking the course is better than

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00:26:08,420 --> 00:26:12,080
before taking the course. So X in UD is positive.

287
00:26:19,090 --> 00:26:27,510
T is D bar minus 0 divided by SD over square root

288
00:26:27,510 --> 00:26:41,090
of A. D bar is 50 divided by 65 divided

289
00:26:41,090 --> 00:26:54,490
by Square root of 8. So 50 divided by square

290
00:26:54,490 --> 00:26:57,910
root of 8, 2.17.

291
00:27:04,070 --> 00:27:09,570
Now Yumi used the critical value approach. So my

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00:27:09,570 --> 00:27:10,930
critical value is T alpha.

293
00:27:13,680 --> 00:27:20,140
And degrees of freedom is 7. It's upper 10. So

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00:27:20,140 --> 00:27:27,300
it's plus. So it's T alpha 0, 5. And DF is 7,

295
00:27:27,320 --> 00:27:33,820
because N equals 8. Now by using the table, at 7

296
00:27:33,820 --> 00:27:34,680
degrees of freedom,

297
00:27:38,220 --> 00:27:39,340
so at 7,

298
00:27:53,560 --> 00:28:03,380
So my T value is greater than the

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00:28:03,380 --> 00:28:07,020
critical region, so we reject the null hypothesis.

300
00:28:10,740 --> 00:28:17,700
The rejection region starts from 1.9895 and this

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00:28:17,700 --> 00:28:24,800
value actually greater than 1.8. So since it falls

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00:28:24,800 --> 00:28:30,320
in the rejection region, then we reject the null

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00:28:30,320 --> 00:28:35,060
hypothesis. It means that taking the course,

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00:28:36,370 --> 00:28:39,690
improves your score. So we have sufficient

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00:28:39,690 --> 00:28:43,010
evidence to support the alternative hypothesis.

306
00:28:44,330 --> 00:28:50,650
That's for number three. The other part, the other

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00:28:50,650 --> 00:28:51,130
part.

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00:28:54,290 --> 00:28:58,550
A statistician selected a sample of 16 receivable

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00:28:58,550 --> 00:29:03,530
accounts. He reported that the sample information

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00:29:04,690 --> 00:29:07,790
indicated the mean of the population ranges from

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00:29:07,790 --> 00:29:12,730
these two values. So we have lower and upper

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00:29:12,730 --> 00:29:21,910
limits, which are given by 4739.

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00:29:36,500 --> 00:29:42,400
So the mean of the population ranges between these

314
00:29:42,400 --> 00:29:47,880
two values. And in addition to that, we have

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00:29:47,880 --> 00:29:55,920
information about the sample standard deviation is

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00:29:55,920 --> 00:29:56,340
400.

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00:29:59,500 --> 00:30:03,260
The statistician neglected to report what

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00:30:03,260 --> 00:30:07,440
confidence level he had used. So we don't know C

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00:30:07,440 --> 00:30:14,180
level. So C level is unknown, which actually is 1

320
00:30:14,180 --> 00:30:14,760
minus alpha.

321
00:30:20,980 --> 00:30:25,360
Based on the above information, what's the

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00:30:25,360 --> 00:30:28,380
confidence level? So we are looking for C level.

323
00:30:29,380 --> 00:30:34,160
Now just keep in mind the confidence interval is

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00:30:34,160 --> 00:30:38,200
given and we are looking for C level.

325
00:30:42,920 --> 00:30:46,600
So this area actually is alpha over 2 and other

326
00:30:46,600 --> 00:30:49,940
one is alpha over 2, so the area between is 1

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00:30:49,940 --> 00:30:50,440
minus alpha.

328
00:30:53,340 --> 00:30:58,620
Now since the sample size equal

329
00:31:01,950 --> 00:31:10,010
16, N equals 16, so N equals 16, so your

330
00:31:10,010 --> 00:31:12,490
confidence interval should be X bar plus or minus

331
00:31:12,490 --> 00:31:14,610
T, S over root N.

332
00:31:19,350 --> 00:31:26,390
Now, C level can be determined by T, and we know

333
00:31:26,390 --> 00:31:28,130
that this quantity,

334
00:31:30,730 --> 00:31:36,970
represents the margin of error. So, E equals TS

335
00:31:36,970 --> 00:31:42,950
over root N. Now, since the confidence interval is

336
00:31:42,950 --> 00:31:50,270
given, we know from previous chapters that the

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00:31:50,270 --> 00:31:53,970
margin equals the difference between upper and

338
00:31:53,970 --> 00:31:59,560
lower divided by two. So, half distance of lower

339
00:31:59,560 --> 00:32:06,320
and upper gives the margin. So that will give 260

340
00:32:06,320 --> 00:32:17,620
.2. So that's E. So now E is known to be 260.2

341
00:32:17,620 --> 00:32:24,320
equals to S is given by 400 and N is 16.

342
00:32:26,800 --> 00:32:29,420
Now, simple calculation will give the value of T,

343
00:32:30,060 --> 00:32:31,340
which is the critical value.

344
00:32:35,280 --> 00:32:38,160
So, my T equals 2.60.

345
00:32:41,960 --> 00:32:47,220
Actually, this is T alpha over 2. Now, the value

346
00:32:47,220 --> 00:32:52,400
of the critical value is known to be 2.602. What's

347
00:32:52,400 --> 00:32:56,520
the corresponding alpha over 2? Now look at the

348
00:32:56,520 --> 00:32:59,660
table, at 15 degrees of freedom,

349
00:33:02,720 --> 00:33:10,680
look at 15, at this value 2.602, at this value.

350
00:33:12,640 --> 00:33:19,880
So, 15 degrees of freedom, 2.602, so the

351
00:33:19,880 --> 00:33:21,940
corresponding alpha over 2, not alpha.

352
00:33:24,610 --> 00:33:31,830
it's 1% so my alpha over 2 is

353
00:33:31,830 --> 00:33:43,110
1% so alpha is 2% so the confidence level is 1

354
00:33:43,110 --> 00:33:50,510
minus alpha so 1 minus alpha is 90% so c level is

355
00:33:50,510 --> 00:33:59,410
98% so that's level or the confidence level. So

356
00:33:59,410 --> 00:34:03,990
again, maybe this is a tricky question.

357
00:34:07,330 --> 00:34:10,530
But at least you know that if the confidence

358
00:34:10,530 --> 00:34:15,270
interval is given, you can determine the margin of

359
00:34:15,270 --> 00:34:18,930
error by the difference between lower and upper

360
00:34:18,930 --> 00:34:23,310
divided by two. Then we know this term represents

361
00:34:23,310 --> 00:34:27,150
this margin. So by using this equation, we can

362
00:34:27,150 --> 00:34:29,770
compute the value of T, I mean the critical value.

363
00:34:30,670 --> 00:34:35,290
So since the critical value is given or is

364
00:34:35,290 --> 00:34:38,590
computed, we can determine the corresponding alpha

365
00:34:38,590 --> 00:34:45,390
over 2. So alpha over 2 is 1%. So your alpha is

366
00:34:45,390 --> 00:34:51,710
2%. So my C level is 98%. That's

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00:34:51,710 --> 00:34:56,180
all. Any questions? We're done, Muhammad.