Question

for the hypothesis test h0:μ=10 against h1:μ<10 with variance unknown and n=10, find the best approximation for the p-value for the test statistic

Answer 1

To calculate the p-value for a hypothesis test with an unknown variance and a small sample size (n=10), use the t-distribution with 9 degrees of freedom. An exact p-value can be determined if the test statistic value is known.

Step-by-step explanation:

For the hypothesis test H0: μ = 10 against H1: μ < 10 with an unknown variance and a sample size of n=10, the appropriate distribution to use is the t-distribution. This is because the sample size is small (< 30) and the population variance is not known. When calculating the p-value for a given test statistic using the t-distribution, one must consider degrees of freedom, which in this case would be n - 1, or 9. The p-value is the probability of obtaining a result at least as extreme as that observed, given that the null hypothesis is true.

Without a test statistic value provided, it's not possible to give the exact p-value. However, if we have the test statistic, we would consult a t-distribution table or use statistical software or a calculator to find the p-value. For example, if the test statistic was -2.08 on a t-distribution with 9 degrees of freedom, the p-value would be found by looking up this value in a t-table or using a calculator's functionality to determine the probability.

Answer 2

The best approximation for the p-value for the test statistic is 0.005775

Finding the best approximation for the p-value for the test statistic

To find the approximation for the p-value, we need to calculate the t-statistic

And then use it to determine the corresponding p-value using a t-distribution table

The t-statistic is calculated as:

t = (x - μ) / (s / √n)

Where:

• x is the sample mean
• μ is the hypothesized mean (10 in this case)
• s is the sample standard deviation
• n is the sample size (10 in this case)

Assume the sample mean is 8 and sample standard deviation (s) is 2, we have

t = (8 - 10) / (2 / √10) = -3.16

Since H1 is one-tailed (μ < 10), we need to find the area to the left of the t-statistic in the t-distribution table with degrees of freedom (df) = n - 1 = 10 - 1 = 9.

Using a t-distribution table, we find that the p-value is approximately 0.005775