# Test of hypothesis involving the population mean 𝝁 when the variance is known

• MHB
• bunnypatotie
In summary, we used the one-sample z-test to determine if the student's belief about the mean score of Grade 11 students was valid, and the results showed that it was indeed lower than the announced mean score.
bunnypatotie
1. The Head of the Mathematics Department announced that the mean score of Grade 11 students in the second periodical test in Statistics was 89, and the standard deviation was 12. One student believed that the mean score was less than this, randomly selected 34 students, computed their mean score, and obtained a mean score of 85. At 0.01 level of significance, test the student’s belief.

- I'm confused on what test statistic to use. I think z-statistic and CLT are both possible for this problem.

Answer:We can use the one-sample z-test to determine if the student's belief is valid. The null hypothesis is that the mean score of Grade 11 students in the second periodical test in Statistics is equal to 89. H0: μ = 89The alternative hypothesis is that the mean score is less than 89.Ha: μ < 89The test statistic is calculated as follows:z = (85 - 89)/(12/√34) = -2.36The critical value at a 0.01 level of significance is -2.33. Since the test statistic is less than the critical value, we reject the null hypothesis and conclude that the student's belief that the mean score was less than 89 is supported at the 0.01 level of significance.

## What is a hypothesis test involving the population mean 𝝁 when the variance is known?

A hypothesis test involving the population mean 𝝁 when the variance is known is a statistical method used to determine whether a sample mean is significantly different from a hypothesized population mean, when the variance of the population is already known. This test helps researchers make inferences about a population based on a sample.

## What is the null hypothesis in a test involving the population mean 𝝁 when the variance is known?

The null hypothesis in a test involving the population mean 𝝁 when the variance is known is that there is no significant difference between the sample mean and the hypothesized population mean. This means that any observed differences between the two can be attributed to chance or sampling error.

## What is the alternative hypothesis in a test involving the population mean 𝝁 when the variance is known?

The alternative hypothesis in a test involving the population mean 𝝁 when the variance is known is that there is a significant difference between the sample mean and the hypothesized population mean. This means that any observed differences between the two cannot be explained by chance or sampling error.

## What is a significance level in a test involving the population mean 𝝁 when the variance is known?

A significance level in a test involving the population mean 𝝁 when the variance is known is the predetermined probability at which the null hypothesis will be rejected. It is typically set at 0.05 or 0.01, meaning that there is a 5% or 1% chance, respectively, of rejecting the null hypothesis when it is actually true.

## What is a p-value in a test involving the population mean 𝝁 when the variance is known?

A p-value in a test involving the population mean 𝝁 when the variance is known is the probability of obtaining a sample mean at least as extreme as the one observed, assuming the null hypothesis is true. It is used to determine whether the null hypothesis should be rejected or not. A p-value less than the significance level indicates that the null hypothesis should be rejected.

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