Chapter 10 project for probability and statistics

MAT 133-Chpt. 10:Hypothesis Testing Name_________________________ Project and Chapter 10 Exam Date__________________________ To complete this assignment, you will do 6 hypothesis testing problems. • You will do each of the problems on your own notebook paper that is labeled with your name and each problem number. • The assignment must be completed and submitted as a PDF (using CamScanner, Genius Scan, etc) to JetNet, by Friday, December 11, 2020. • Please Note: This due date cannot be changed. • Each of the problems will be completed using the P-Value Method for Hypothesis Testing. • To receive credit, each problem must be completed using the following steps: Preliminary Steps: First, we must verify the requirements to perform the hypothesis test: 1. Random: 2. Independent: 3. Normal? P-Value Method: Step 1: Determine the null and alternate hypotheses. (Note: 0p is the assumed value of the population proportion.) 0H = 1H = Step 2: The level of significance is: Step 3: Find the test statistic: The sample proportion is: ˆ x p n = = The test statistic is: ( ) 0 0 0 0 ˆ 1 p p z p p n − = = − Note: Circle the correct response. This is a left-tailed two-tailed right-tailed test. Step 4: Determine the P-Value using the test output on the calculator. i. Show(write down) what you entered in the calculator and the results it gave you. ii. Label the values of the test statistic and p-value on the diagram and shade the appropriate area. Step 5: Reject 0H if the P value−  (P-value is less than alpha) Be sure to show how you decided what decision to make. Step 6: State the conclusion. 3 P-Value Method for Hypothesis Testing Hypothesis Testing Using the P-Value Approach If the probability of getting a sample statistic as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis. Definition A P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true. Put another way, the P-value is the likelihood or probability that a sample will result in a statistic such as the one obtained if the null hypothesis is true. Solving Hypothesis-Testing Problems (P-Value Method) Step 1 State the hypotheses and identify the claim. Step 2 Compute the test value. Step 3 Find the P-value. Step 4 Make the decision. Step 5 Summarize the results. If P value−  , reject the null hypothesis. If P value−  , do not reject the null hypothesis. 4 Note: Assumed Standard Error of the Sample Proportion ( ) ( )0 01ˆ p p SE p n −  Testing Hypotheses Regarding a Population Proportion, p – 1-propZ-Test: (10.2) Requirements to Conduct Test: • Random Sample: Simple random sample (SRS) of size n • Independent Sample: If sampling without replacement, 0.05n N • Normal: ( )0 01 10np p−  Step 1: Determine the null and alternative hypotheses. (Note: 0p is the assumed value of the population proportion) Two-tailed Left-tailed Right-tailed 0 0 1 0 : : H p p H p p =  0 0 1 0 : : H p p H p p =  0 0 1 0 : : H p p H p p =  Step 2: Select a level of significance α based on the seriousness of making a type I error. Step 3: Find the test statistic ( ) 0 0 0 0 ˆ 1 p p z p p n − = − Classical Approach P-value Approach Step 4: Determine the critical z-value(s) (Use invNorm or bottom row of T-Table) Two-tailed Left-tailed Right-tailed Step 4: Determine the P-value using the test output on the calculator Two-tailed Left-tailed Right-tailed Step 5: Compare the critical value(s) with the test statistic. Two-tailed Left-tailed Right-tailed Reject 0H if either 0 /2z z − or 0 /2z z Reject 0H if 0z z − Reject 0H if 0z z Step 5: Reject 0H if the P-value < α Step 6: State the conclusion /2/2 z-z/2 /2  -z  z P-value |z | 0 -|z | 0 P-value z0 P-value z0 5 Hypothesis Test for a Single Proportion, p – Ti-84: 1. Press to highlight the Stat TESTS menu. 2. Press to select 5:1-PropZTest [See Left Picture] 3. In the 1-PropZTest menu: [See Right Picture] a. 0 :p Enter the value of the population proportion stated in the null hypothesis b. x: Enter the number of successes, x (Note: x must be a whole number. You may need to calculate x by using the formula for p̂ and rounding!!) c. n: Enter the sample size, n d. prop: Move the cursor (with the or key) to highlight the correct alternative hypothesis and press to select it e. Calculate: After the items above are entered, move down to highlight Calculate and press 6 1) Testing a Hypothesis about a Population Proportion: Large Sample Size In 1997, 46% of Americans said they did not trust the media “when it comes to reporting the news fully, accurately and fairly”. • In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. • Is there enough evidence to support the claim that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997? Test the hypothesis using the P-value method at the 0.05 = level of significance, showing all steps. 2) People Who Are Trying to Avoid Trans Fats A dietitian claims that 60% of people are trying to avoid trans-fats in their diets. • She randomly selected 200 people and found that 128 people stated that they were trying to avoid trans-fats in their diets. • Is there enough evidence to show that the number of people trying to avoid trans-fats in their diet is different than 60%? Test the hypothesis using the P-value method at the 0.05 = level of significance, showing all steps. 3) Survey on Call-Waiting Service A telephone company representative estimates that 40% of its customers have call- waiting service. • To test this hypothesis, she selected a random sample of 100 customers and found that 37% had call waiting. • Is there enough evidence to show that the number of customers with call-waiting is less than 40%? Test the hypothesis using the P-value method at the 0.01 = level of significance, showing all steps. 7 Testing Hypotheses Regarding a Population Mean, μ - T-test: (10.3) Requirements to Conduct Test: • Random Sample: Simple random sample (SRS) of size n • Independent Sample: If sampling without replacement, 0.05n N • Normal: Either population distribution is normal (given/graph) or 30n  . No outliers. Step 1. State the null and alternative hypotheses Two-tailed Left-tailed Right-tailed 0 0 1 0 : : H H     =  0 0 1 0 : : H H     =  0 0 1 0 : : H H     =  Step 2. Select a level of significance α based on the seriousness of making a type I error. Step 3. Find the test statistic: 00 x t s n − = Classical Approach: P-value Approach: Step 4. Determine the critical t-value(s) using invT or the T-Table (df = n-1) Two-tailed Left-tailed Right-tailed Step 4. Determine the P-value using the test output on the calculator Two-tailed Left-tailed Right-tailed Step 5. Compare the critical value(s) with the test statistic. Two-tailed Left-tailed Right-tailed Reject 0H if either 0 /2 −t t or 0 /2t t Reject 0H if 0  −t t Reject 0H if 0 t t Step 5. Reject 0H if the P-value < α Step 6. State the conclusion Note: Assumed Standard Error of the Sample Mean ( ) s SE x n  /2/2 t-t/2 /2  -t  t P-value |t |0-|t |0 P-value t0 P-value t0 8 Hypothesis Test for a Single Mean, μ, from RAW Data – Ti-84: 1. Press select 1: EDIT. 2. Enter the raw data in list L1. 3. Press to highlight the Stat TESTS menu. 4. Press to select 2:T-Test [See 1st Picture] 5. In the T-Test menu: [See 2nd Picture] a. Inpt: Press while the cursor highlights “Data” to select Data. b. 0 : Enter the value of the population mean stated in the null hypothesis. c. List: Press to enter list L1, if necessary d. Freq: Enter 1. e. : Move the cursor (with the or key) to highlight the correct alternative hypothesis and press to select it f. Calculate: After the items above are entered, move down to highlight Calculate and press Hypothesis Test for a Single Mean, μ, from Summary Statistics – Ti-84: 1. Press to highlight the Stat TESTS menu. 2. Press to select 2:T-Test [See 1st Picture] 3. In the T-Test menu: [See 2nd Picture] a. Inpt: Press while the cursor highlights “Stats” to select Stats. b. 0 : Enter the value of the population mean stated in the null hypothesis. c. :x Enter the given sample mean. d. :Sx Enter the given sample standard deviation. e. : Move the cursor (with the or key) to highlight the correct alternative hypothesis and press to select it f. Calculate: After the items above are entered, move down to highlight Calculate and press 9 1) Hypothesis about a Population Mean: Large Sample • The mean height of American males is 69.5 inches. • The heights of the 43 male U.S. presidents (Washington through Obama) have a mean 70.78 inches and a standard deviation of 2.77 inches. • Treating the 43 presidents as a simple random sample, is there enough evidence to suggest that U.S. presidents are taller than the average American male. Test the hypothesis using the P-value method at the 0.05 = level of significance, showing all steps. 2) Costs of Men’s Athletic Shoes • A researcher claims that the average cost of men’s athletic shoes is less than $80. • He selects a random sample of 36 pairs of shoes from a catalog and finds the following costs (in dollars). (The costs have been rounded to the nearest dollar.) • Is there enough evidence to support the researcher’s claim that the average cost of men’s athletic shoes is less than $80? • Assume = 19.2s . Test the hypothesis using the P-value method at the = 0.10 level of significance, showing all steps. 3) Cost of Rehabilitation The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24,672. • A researcher thinks the average cost of rehabilitation is different. • To see if the average cost of rehabilitation is different at a particular hospital, the researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is $25,226. • Is there enough evidence to support the researcher’s claim that the average cost of rehabilitation for stroke victims is different from $24,672? • The sample standard deviation is: = $3251s . Test the hypothesis using the P-value method at the = 0.01 level of significance, showing all steps.