Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Math 114, Homework 6

Due: This assignment is due Monday October 18th by 11pm. Submit the assignment on gradescope. Show all your work and give complete explanations of what you are doing. If you use an applet for any question, include a screen shot of the applet. However, remember that a screen shot of an applet does not constitute enough explanation; you must also give a written interpretation of the results obtained using the applet. Write in complete, grammatically correct, sentences.

Section 3.1 # 24 (in part d, describe what you did)

Section 3.2 # 9, 15

Question 1: Gas mileage of new Honda Civics

Recall the test discussed in class to determine if the new Honda Civic gets better mileage than the old Honda Civic. The null and alternative hypotheses of the test are

H_0: μ = 38 and σ = 6.4

versus

H_a: μ > 38

They use as the test statistic, and the test is conducted at the 2% significance level. Suppose now that the researchers want the power of the test when μ = 42 and σ = 6.4 to be 95%. What sample size should they use?

Question 2: Drinking Soda Pop Daily

In a July 2012 Gallup Poll based on a representative sample of 1014 adult Americans, 48% reported drinking at least one glass of soda pop on a typical day. Let π denote the proportion of all adult Americans who drink at least one glass of soda pop on a typical day.

a) Is it plausible, given the data that π = 0.5?

i. Formulate the null and alternative hypotheses you would use to determine this.

ii. It is reasonable to use a normal approximation for the null distribution of  in this test? Justify your answer.

iii. Calculate the p-value. Draw a picture of the distribution of  you are using to find the p-value. Indicate the mean and standard deviation on your picture and shade the area corresponding to the p-value.

iv. What is your conclusion? Is 0.5 plausible? (Use α = 0.1. )

b) Let [πl, πu] be a 90% confidence interval for π. Let’s start by finding πl.

i. What should the p-value be when conducting the test H0: π = πl versus Ha: π ≠ πl?

ii. What is the null distribution of  in this test? Be specific. Draw a picture of the distribution. Label the mean and standard deviation and shade the region whose area is equal to the p-value of the test.

iii. Draw the corresponding picture you get of the standard normal distribution when you standardize. Shade the area whose area is equal to the p-value of the test and find the value on the axis where this region starts in two ways; using the standard normal tables (because you know what the area should be equal to) and as the z-score of a number from the null distribution of . This gives you an equation in πl.

iv. What approximation do we make before solving this equation? Make the approximation and solve the equation to find πl.

c) Work through the steps in part b) again adjusting them appropriately to find πu.

d) Based on the confidence interval you found, is it plausible (at the 10% significance level), that 45% of Americans drink at least one glass of soda pop on a typical day?

Question 3: Left-handed students

Among a random sample of 97 male students at Hope College, 12 were left-handed.

a) Use theory (and not an applet) to determine the quick-and-dirty 95% confidence interval for the proportion of all male students at Hope College who are left-handed.

b) Now determine a 99% confidence interval for this proportion.

c) Give your answer to b) is it plausible (at the 1% significance level) that 10% of male students at Hope College are left-handed?