Question

Extreme cold and hot temperatures are known to affect the operation of electronic components. Winter is approaching and you are interested in the mean minimum temperature that will damage an iPod. As a research project, 9 iPods were tested to see what minimum temperature damaged the electronic component. The damaging temperature averaged 5 ºF with a standard deviation of 3 ºF. Find a 95% confidence interval for the true mean minimum temperature that will damage an iPod. (Disclaimer: Apple recommends that to "keep iPod comfy," the iPod’s battery should be kept between temperatures of 32 ºF and 95 ºF.)If the question requires a test of significance, your solution should clearly show the four steps to the test. In addition, you should show that you have checked that the conditions for inference are met.Step 1: State the null and alternative hypothesis.Step 2: Calculate the test statistic.Step 3: Find the p-value.Step 4: State your conclusion in the context of the problem. (Do not just say "Reject H0" or "Do not reject H0.")If the question requires a confidence interval, be sure to make a statement about the confidence interval in context of the problem.

Answer 1

A 95% confidence interval for the true mean minimum temperature that will damage an iPod is between 2.55°F and 7.45°F

Test statistic(Z) = 2.31

P-value = 0.0104

Explanation:

Step 1

Null hypothesis: The average damaging temperature of nine iPods is 5°F

Alternate hypothesis: The average damaging temperature of nine iPods differs from 5°F

Step 2

Mean=5°F, Sd=3, df=n-1=9-1=8

The t-value corresponding to 8 degrees of freedom and 95% confidence level (5% significance level) is 2.306

Confidence Interval(CI) = (mean + or - t×sd/√n)

CI = (5 + 2.306×3/√8) = 5 + 6.918/2.828= 5+2.45=7.45°F

CI = (5 - 2.306×3/√8) = 5-2.45 = 2.55°F

Z = (sample mean - population mean)/(sd÷√n) = (5-2.55)/(3÷√8) = 2.45/1.061 = 2.31

Step 3

Using the standard distribution table, the cumulative area to the left of Z = 2.31 is 0.9896

P-value = 1 - 0.9896 = 0.0104

Step 4

Conclusion: A 95% confidence interval for the true mean minimum temperature that will damage an iPod is between 2.55°F and 7.45°F