Kind attention Psydag:

A random sample of size 20 from a normal population with sample mean 42 and sample standard deviation of 6 . To test the hypothesis the population mean 44 at 5% level of significance , find the p-value and comment

Clarification:

The sample size is 20 which is less than 30. Z tables cannot be used for this problem. It is given that the distirbution is normal with small sample size and hence t distribution should be used. I need some assistance in finding the P value using t distribution.

Can you please throw some light on this particular point.
Thanks for the response. Mrs. Lakshmi

Dear Mrs. Lakshmi,

To find the p-value using a t-distribution, we need to perform a t-test. The formula for a t-test is:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

In this case, the sample mean is 42, the hypothesized mean is 44, the sample standard deviation is 6, and the sample size is 20. Plugging these values into the formula, we get:

t = (42 - 44) / (6 / sqrt(20))
= -2 / (6 / sqrt(20))

To calculate the p-value, we need to find the probability of observing a t-value as extreme as the one we calculated (in this case, -2) or more extreme, assuming the null hypothesis is true.

Using the t-distribution table or a calculator, we can find the p-value corresponding to the t-value. The p-value conventionally represents the probability of observing a more extreme result than the one observed, assuming the null hypothesis is true.

For a two-tailed test (which is typically used when the alternative hypothesis is simply that the population mean is not equal to 44), we need to find the probability in both tails, and then multiply it by 2.

The calculated t-value of -2 corresponds to a p-value of approximately 0.0322 (assuming a degrees of freedom of 19), which is the probability of observing a result as extreme as -2 or more extreme.

Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis. This means that there is evidence to suggest that the population mean is different from 44.

I hope this explanation helps! Let me know if you have any further questions.

Sincerely,
Explain Bot