(A) An architect estimates that the average height of the buildings of 30 or more stories in Suva is at least 500 feet. A random sample of 12 buildings is selected, and the heights in feet are shown. At =0.025, is there enough Evidence to reject the claim?
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(B)A survey by Men’s Health magazine stated that 40% of men said they used alcohol to reduce stress. At 0.10,test the claim that a random sample of 100 men was selected and 30 said that they used alcohol to reduce stress. Use the P-value method
(A) To determine if there is enough evidence to reject the claim that the average height of buildings of 30 or more stories in Suva is at least 500 feet, we can perform a hypothesis test.
Step 1: State the hypotheses:
The null hypothesis (H0): The average height of buildings of 30 or more stories in Suva is 500 feet or less.
The alternative hypothesis (Ha): The average height of buildings of 30 or more stories in Suva is greater than 500 feet.
Step 2: Set the significance level:
Given α = 0.025, this is the probability of making a Type I error by rejecting the null hypothesis when it is actually true.
Step 3: Calculate the test statistic:
We need to calculate the sample mean (x-bar) and the sample standard deviation (s) from the given data. Then, we can calculate the test statistic using the formula:
t = (x-bar - μ) / (s / √n)
where x-bar is the sample mean, μ is the hypothesized population mean (500 feet), s is the sample standard deviation, and n is the sample size.
Step 4: Calculate the p-value:
Using the test statistic calculated in step 3, we can find the p-value associated with it. The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
Step 5: Make a decision:
Compare the p-value calculated in step 4 with the significance level (α). If the p-value is less than α, we reject the null hypothesis. Otherwise, if the p-value is greater than or equal to α, we fail to reject the null hypothesis.
(B) To test the claim that 40% of men use alcohol to reduce stress, we can use the P-value method.
Step 1: State the hypotheses:
The null hypothesis (H0): The true proportion of men who use alcohol to reduce stress is 40%.
The alternative hypothesis (Ha): The true proportion of men who use alcohol to reduce stress is not 40%.
Step 2: Set the significance level:
Given α = 0.10, this is the probability of making a Type I error by rejecting the null hypothesis when it is actually true.
Step 3: Calculate the test statistic:
We can use the formula for the test statistic for proportions:
z = (p-hat - p0) / √(p0 * (1 - p0) / n)
where p-hat is the sample proportion (30/100 = 0.30), p0 is the hypothesized population proportion (0.40), and n is the sample size (100).
Step 4: Calculate the p-value:
Using the test statistic calculated in step 3, we can find the p-value associated with it. The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
Step 5: Make a decision:
Compare the p-value calculated in step 4 with the significance level (α). If the p-value is less than α, we reject the null hypothesis. Otherwise, if the p-value is greater than or equal to α, we fail to reject the null hypothesis.