A statistics instructor is interested in the ability of...

A statistics instructor is interested in the ability of students to assess the difficulty of a test they have taken. This test was taken by a large group of students, and the average score was 78.5. A random sample of eight students was asked to predict this average score. Their predictions were as follows: $\begin{array}{llllllll}72 & 83 & 78 & 65 & 69 & 77 & 81 & 71\end{array}$ Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a two-sided alternative and a $10 \%$ significance level.

A statistics instructor is interested in the ability of students to assess the difficulty of a test they have taken. This test was taken by a large group of students, and the average score was 78.5. A random sample of eight students was asked to predict this average score. Their predictions were as follows:
$\begin{array}{llllllll}72 & 83 & 78 & 65 & 69 & 77 & 81 & 71\end{array}$
Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a two-sided alternative and a $10 \%$ significance level.

Key Concepts

Two-Tailed Test

A two-tailed test is used when deviations in both directions away from the hypothesized value are considered equally significant. In this type of test, the rejection region is split between both tails of the distribution, and the null hypothesis is rejected if the test statistic falls into either tail.

One-Sample t-Test

A one-sample t-test is used to compare the mean of a sample with a known or hypothesized population mean when the sample size is small and the population standard deviation is unknown. It involves calculating a t-statistic which follows a t-distribution with degrees of freedom equal to the sample size minus one.

Significance Level

The significance level (alpha) is the threshold that determines whether the observed data are sufficiently unlikely under the null hypothesis. It represents the probability of making a Type I error, which is the incorrect rejection of a true null hypothesis. In this context, a 10% significance level means that there is a 10% risk of concluding that a difference exists when there is none.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves setting up a null hypothesis to represent a baseline state (usually a statement of no effect or no difference) and an alternative hypothesis which represents what one is trying to provide evidence for.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) is the statement being tested and is presumed true until evidence indicates otherwise, while the alternative hypothesis (H1) is what the researcher aims to support. In the context of testing a mean, the null hypothesis typically states that the population mean is equal to a specific value, whereas the alternative hypothesis posits that the population mean differs from that value.

Please give Ace some feedback