A random sample of n= 12 is drawn from a population that is normally distributed, and the sample variance is s©÷ = 19.3. Use ¥á = 0.025 in testing

¥Ç₀ : ¥ò©÷ ¡ 9.4 versus ¥Ç©û : ¥ò©÷ > 9.4

Yadira, Marianne, Laura, Vanessa, or whoever -- please don't keep switching your name. We know that these statistics problems have been posted from the same computer -- and presumably by the same person.

If you are in fact different persons, shouldn't you be helping each other? In any case, together with your course notes, there are sufficient examples for you to understand the subject.

I suggest you work out your next problems and post the results for checking.

To test the hypothesis ¥Ç₀ : ¥ò©÷ ¡ 9.4 versus ¥Ç©û : ¥ò©÷ > 9.4, we can use the one-sample t-test because the population is assumed to be normally distributed and we know the sample variance.

Here are the steps to perform the hypothesis test:

Step 1: State the null and alternative hypotheses:
Null Hypothesis (¥Ç₀): ¥ò©÷ = 9.4
Alternative Hypothesis (¥Ç©û): ¥ò©÷ > 9.4

Step 2: Determine the significance level (¥á) and the corresponding critical value:
Given that ¥á = 0.025, we want to find the critical value corresponding to this significance level from the t-distribution. Since the alternative hypothesis is one-sided (¥Ç©û: ¥ò©÷ > 9.4), we will use the upper tail of the t-distribution.

Using a t-table or calculator, we find that the critical value for a one-sided test with ¥á = 0.025 and degree of freedom (df = n-1) of 11 is approximately 2.228.

Step 3: Calculate the test statistic:
The test statistic for a one-sample t-test is given by:
t = (x̄ - µ₀) / (s / √n)

Where:
x̄ is the sample mean
µ₀ is the hypothesized population mean (9.4 in this case)
s is the sample standard deviation (square root of the sample variance, which is 19.3)
n is the sample size (12 in this case)

Given the sample variance s©÷ = 19.3, we take the square root to find the sample standard deviation s ≈ √19.3 ≈ 4.394.

Using the given data: n = 12 and s ≈ 4.394, we can calculate the test statistic:
t = (x̄ - µ₀) / (s / √n) = (x̄ - 9.4) / (4.394 / √12)

Step 4: Calculate the p-value:
To calculate the p-value, we need to find the probability of getting a test statistic as extreme as the one observed (or more extreme) if the null hypothesis is true.

p-value = P(t > calculated test statistic)

By finding this probability from the t-distribution with n-1 degrees of freedom, we can determine the p-value.

Step 5: Make a decision:
If the p-value is less than the significance level (¥á), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

I can assist you in solving the calculations if you provide the sample mean (x̄) for the dataset.