# Directions: For both studies, complete all steps of the 4-step method for hypothesis tests/tests of significance. Round all critical values (z*) to 2 decimal places. In addition, round p-values...

*Directions:*

For both studies, complete all steps of the ** 4-step method for hypothesis tests/tests of significance**. Round all

**(z*) to**

*critical values***. In addition, round**

*2 decimal places***to**

*p-values***and all**

*4 decimal places***values to**

*other***.**

*2 decimal places* *Study 1*

Ratings often overstate true fuel economy. A car owner keeps careful records of the gas mileage of her new Civic hybrid for every 100 mile increment for 3000 miles of highway driving (30 observations total). Her result is an average of 47.2 miles per gallon (mpg). She wonders whether the data show that her true long-term average highway mileage is less than 51 mpg. Find the answer with a significance level of 0.01. Assume a population standard deviation of 10 mpg.

*Study 2*

Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a month at a high temperature to imitate the effect of four months’ storage at room temperature. Each taster scores the cola again after storage. Our data are the differences in the tasters’ scores. The bigger these differences, the larger the loss of sweetness. Suppose we know that for any cola, the sweetness loss scores vary from taster to taster with a normal distribution with standard deviation of 1. For a certain brand name diet cola, 16 trained tasters had a mean sweetness loss of 1.02. For a certain generic diet cola, another 14 trained testers had a mean sweetness loss of 1.15. Determine whether the mean sweetness scores for the two colas differ.

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### 1 Answer

The gas mileage of the Civic hybrid is given as 51 mpg with a standard deviation of 10 mpg. The population variance is 100. She measures the mileage of her car every 100 miles for 3000 miles. This gives 30 samples. Her sample average is 47.2.

Using the values provided:

`mu` = 51

v = 100

M = 47.2

n = 30

The Z score value is `(M - mu)/sqrt(v/n)` = -2.08135

This Z score gives a p value of 0.03752. The result is not significant at p < 0.01.

When the two colas are tasted by the group of trained testers, a brand name cola gets a score of 1.02 with a sample size 16. And the generic soda with a sample size 14 gets a score of 1.15. The score given by each taster lies on a normal distribution curve with standard deviation 1.

The range of the mean sweetness of the brand name cola is 0.38 to 1.66 with a 99% desired confidence level.

For the same parameters, the generic cola has a mean sweetness between 0.46 to 1.84