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Stats Problem: Test the null hypothesis that the mean run time is at least 300 minutes?

• Common information 1-4: An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. Suppose a simple random sample of 49 engines is tested. The engines run for an average of 295 minutes, assume the population standard deviation is 20 minutes. Test the null hypothesis that the mean run time is at least 300 minutes. Use a left sided test. Use a 0.025 level of significance. 1] Find the critical value for this test. a. 1.645 b. -1.645 c. -1.96 d. -1.684 e. None of the above 2] Find the test statistic a. -1.645 b. 1.768 c. -1.786 d. -1.75 e. None of the above 3] Find the p-value for this test. a. .0802 b. -1.77 c. .0768 d. .0401 e. None of the above 4] What is the correct decision? a. Reject Ho in favor of Ha. b. There is insufficient evidence to reject Ho c. Do not reject Ho since P-value <alpha d. Reject Ho since P-value > significance level e. None of the above Please show all work Thank you

• Answer:

  1. The null hypothesis in this case is that it runs for 300 minutes, alternative hypothesis is that it does not... To find the critical value, you look at the normal distribution table and find the corresponding value for the level of significance you are looking for, in this case it's 0.025. On the table, that corresponds to a critical value of -1.96, so the answer is c. 2. First you find the standard error for the sample size. The population standard deviation is 20 and the sample size is 49, so 20/sqrt(49) = 2.857 for the sample mean standard error. Now, to find the actual Test Statistic, you do sample mean - hypothesized mean over standard error: (295 - 300) / 2.857 = -1.75 so the answer is d. 3. The p-value will tell us the probability that the test statistic (-1.75) is less than or equal to -1.75. Using the normal distribution table, the corresponding value for -1.75 is 0.0401. so the answer is d, 0.0401 4. The rule is to reject if the p-value is less than or equal to alpha (level of significance)... in this case, the p-value of -1.75 is less than 0.025, so you should reject the null hypothesis