10. A teacher gives 200 students a study guide for a test and the average score was 90 with a standard deviation of 6. She did not give the other 200 students a study guide and their average score was 70 with a standard deviation of 8. Find the critical value to determine whether or not the study guide helped students to increase their test score.
To find the critical value in this scenario, we need to perform a hypothesis test comparing the two groups: the students who received the study guide and the students who did not.
The first step is to state the null hypothesis (H0) and the alternative hypothesis (Ha).
Null hypothesis (H0): The study guide did not help increase the test scores.
Alternative hypothesis (Ha): The study guide did help increase the test scores.
Next, we need to choose the significance level (α), which is the probability of rejecting the null hypothesis when it is true. Let's choose a significance level of α = 0.05, which is commonly used in statistical analysis.
Now, we can calculate the test statistic to determine the critical value. In this case, we'll be using an independent samples t-test because we are comparing the means of two independent groups.
The formula to compute the test statistic (t) is:
t = (X1 - X2) / sqrt((s1^2 / n1) + (s2^2 / n2))
Where:
X1 and X2 are the means of the two groups,
s1 and s2 are the standard deviations of the two groups,
n1 and n2 are the sample sizes of the two groups.
Given:
X1 = 90 (mean of the study guide group),
s1 = 6 (standard deviation of the study guide group),
n1 = 200 (sample size of the study guide group),
X2 = 70 (mean of the non-study guide group),
s2 = 8 (standard deviation of the non-study guide group),
n2 = 200 (sample size of the non-study guide group).
Substituting the values into the formula, we have:
t = (90 - 70) / sqrt((6^2 / 200) + (8^2 / 200))
Calculating this expression will yield the value of t. Once we have the value of t, we can determine the critical value by consulting the t-distribution table or using statistical software or calculators.
The critical value represents the threshold above which we reject the null hypothesis. If the calculated t-value exceeds the critical value, we reject the null hypothesis, indicating that the study guide did help increase the test scores.
Note: The degrees of freedom for this test are equal to the sum of the individual degrees of freedom for each group, which is (n1 - 1) + (n2 - 1).
After finding the critical value from the t-distribution table or a statistical calculator, you can compare it to the calculated t-value to determine whether the study guide significantly helped increase the test scores.