Given a sample size of 36, with sample mean 400 and sample standard deviation 85, we perform the following hypothesis test, at the 95% level of confidence: Ho: ƒÊ = 450, Ha: ƒÊ �‚ 450. What is the question and what is the answer?

The question is: "Based on the given sample data, can we conclude that the population mean is significantly different from 450?"

To answer this question, we can perform a hypothesis test. The null hypothesis (Ho) states that the population mean (µ) is equal to 450, and the alternative hypothesis (Ha) states that the population mean is not equal to 450.

To perform the hypothesis test, we can use the t-test since the population standard deviation is unknown. Here are the steps to find the answer:

Step 1: Define the hypotheses:
Ho: µ = 450 (null hypothesis)
Ha: µ ≠ 450 (alternative hypothesis)

Step 2: Determine the significance level, α:
The significance level is given as 95% or 0.95 (confidence level). Therefore, α = 1 - 0.95 = 0.05.

Step 3: Compute the test statistic:
The test statistic for a t-test is calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / √n)

Substituting the values into the formula, we get:
t = (400 - 450) / (85 / √36)
t = -50 / (85 / 6)
t = -50 * 6 / 85
t ≈ -3.529

Step 4: Determine the critical value(s):
Since the alternative hypothesis is two-tailed (µ ≠ 450), we need to find the critical values in both tails of the t-distribution.

With a sample size of 36 and a desired 95% confidence level, the critical value can be found using a t-distribution table or a statistical software. For a two-tailed test and α = 0.05, the critical value is approximately ±2.030.

Step 5: Make a decision:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the absolute value of the test statistic (3.529) is greater than the critical value (2.030). Therefore, we reject the null hypothesis.

Step 6: Draw a conclusion:
Based on the sample data, we have enough evidence to conclude that the population mean is significantly different from 450 at the 95% level of confidence.