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1 BA 275 Quantitative Business Methods Statistical Inference for Small Sample Size Statistical Inference for Two-Sample Problems Agenda

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2 Quiz #4: Question 1 Past experience indicates that the monthly long-distance telephone bill is normally distributed with a mean of $17.85 and a (population) standard deviation $3.87. After an advertising campaign aimed at increasing long-distance telephone usage, a random sample of 25 household bills was taken. You are concerned whether the campaign was successful, and would like to perform a test to find out. 1. What are the null and the alternative hypotheses? 2. If the sample mean turns out to be $15, do you reject the null hypothesis? Why or why not? Assume = 5%. 3. If the sample mean turns out to be $29.13, do you reject the null hypothesis? Why or why not? Assume = 5%. 4. Finally, the actual sample mean in your sample is $ Do you reject the null hypothesis? Was the campaign successful? Assume = 5%.

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3 Quiz #4: Question 2 The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. The mean score for U.S. college students is about 115, and the standard deviation is about 30. A teacher suspects that older students (30 years or older) have better attitudes toward school and wishes to test H0: = 115 vs. Ha: > 115. To test the hypothesis, she decides to use a sample of n = 25, and to reject the null hypothesis H0 if the sample mean > Find the probability of a Type I error. 2. Find the probability of a Type II error when = Find the power of the test when = Suppose the sample mean turns out to be Find the p-value of the test. 5. Construct a 95% confidence interval for the mean SSHA score for older students. Write your final answer in the following format: ( point estimate ) ± ( margin of error )

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4 Central Limit Theorem (CLT) is unknown but n is large

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5 Central Limit Theorem (CLT) is unknown and n is small

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6 T distribution T distribution with degrees of freedom 5 vs. Normal(0, 1)

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7 Example 1 (Example 3 from ) A random sample of 10 one-bedroom apartments (Ouch, a small sample) from your local newspaper gives a sample mean of $541.5 and sample standard deviation of $ Assume = 5%. Q1. Does the sample give good reason to believe that the mean rent of all advertised apartments is greater than $500 per month? (Need H0 and Ha, rejection region and conclusion.) Q2. Find the p-value. Q3. Construct a 95% confidence interval for the mean rent of all advertised apartments. Q4. What assumption is necessary to answer Q1-Q3.

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8 Example 2 A bank wonders whether omitting the annual credit card fee for customers who charge at least $2400 in a year would increase the amount charged on its credit card the following year. A random sample of 51 customers is chosen to see if the mean amount charged increases from the previous year under the no- fee offer. The mean increase is $342 and the standard deviation is $108. Q1. Let = 5%. State H0 and Ha and carry out a t test. Approximate the p-value. Q2. Give a 95% confidence interval for the mean amount of the increase. Q3. Suppose that the bank wanted to be quite certain of detecting a mean increase of = $100 in the amount charged. Is n = 51 enough to detect the increase? SG Demo

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9 Two-Sample Inference on 1 – 2 Population #1Population #2 Sample #2Sample #1

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10 Example 3 U.S. SalesJapan Sales Sample size3050 Sample mean$14,545$15,243 Sample Std.$ 1,989$ 1,842 Q1. Do we have enough evidence to claim that the auto retail price in Japan is higher? Q2. If so, by how much?

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11 Example 4 Do government employees take longer coffee breaks than private sector workers?

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12 Answer Key Example 1: see the slides from Example 2: Q1. H0: = 0 vs. Ha: > 0. Given = 5% and df = n – 1 = 50, the rejection region is defined as: Reject H0 if t > (or if the sample mean > ) Since, we should reject H0. Q ± x 108 / SQRT(51) Q3. Power = P( detecting a mean increase of $100 ) = P( being able to reject H0 when true = $100 ) = P( the sample mean > when = $100 ) = Note that came from the rejection region in Q1.

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13 T Table (Table D)

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