7.5.1: Exercises
- Page ID
- 49040
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- Jude was curious if the automated machine at his restaurant was filling drinks with the proper amount. He filled a sample of 20 drinks to test \(H_0: \mu=530 \mathrm{~mL}\) versus \(H_a: \mu \neq 530\) where \(\mu\) is the mean filling amount. The drinks in the sample contained a mean amount of 528 mL with a standard deviation of 4 mL. These results produced a test statistic of \(T=-2.236\) and a P-value of approximately 0.038.
- If the significance level is 5%, what can we conclude about the null hypothesis?
- If the significance level is 5%, what can we conclude about the alternative hypothesis?
- Explain why you made the choices above.
- State the conclusion in context.
- What type of error might have occurred based on the above conclusion?
- Describe the error in context for this example.
- If the significance level is 5%, what can we conclude about the null hypothesis?
- A quality control engineer is testing the battery life of a new smartphone. The company is advertising that the battery lasts 24 hours on a full charge, but the engineer suspects that the battery life is actually less than that. They take a random sample of 30 of these phones to test \(H_0: \mu=24\) versus \(H_a: \mu<24\) where \(\mu\) is the mean battery life of these phones. The sample data had a mean of 21 hours and a standard deviation of 16 hours. These results produced a test statistic of \(T\approx-1.03\) and a P-value of approximately 0.156.
- If the significance level is 5%, state the conclusion in context.
- What type of error might have occurred based on the conclusion above?
- Describe the error in context for this example.
- Suppose we increase the sample size to 300 and find the same sample mean and sample standard deviation.
- Compute the new T-score.
- Is the sample mean statistically significant? Explain why or why not.
- Is the sample mean significant in a practical or real-world sense? Explain why or why not.
- Compute the new T-score.
- If the significance level is 5%, state the conclusion in context.
- According to a report from the United States Environmental Protection Agency, burning one gallon of gasoline typically emits about 8.9 kg of \(\mathrm{CO}_2\). A fuel company wants to test a new type of gasoline designed to have lower \(\mathrm{CO}_2\) emissions. Here are their hypotheses: \(H_0: \mu=8.9 \mathrm{~kg}\), \(H_a: \mu<8.9 \mathrm{~kg}\) (where \(\mu\) is the mean amount of \(\mathrm{CO}_2\) emitted by burning one gallon of this new gasoline).
- Describe the type I error in context for this example.
- Describe the type II error in context for this example.
- Suppose Norton writes the following conclusion to the above hypothesis test:
The P-value is 0.04 which is greater than the level of significance of 1%. Therefore, we accept the null hypothesis and reject the alternative hypothesis. We have proved that the mean amount of \(\mathrm{CO}_2\) emitted by burning one gallon of this new gasoline is not less than 8.9 kg.
Improve Norton’s conclusion.
- Describe the type I error in context for this example.