Test the hypothesis that for the population of students in this professor's classes each additional homework problem complete leads to an increase of at least 2 points on the quiz versus the alternative that the increase is less than 2 points. Allow for 5% type 1 error.
Quiz Score Number of HW problems
1 1
2 1
2 2
3 2
5 3
4 3
7 4
6 4
8 5
10 5
To test this hypothesis, we can use a hypothesis testing framework. Here are the steps to follow:
Step 1: Define the null and alternative hypotheses:
Null hypothesis (H₀): Each additional homework problem completed does not lead to an increase of at least 2 points on the quiz.
Alternative hypothesis (H₁): Each additional homework problem completed leads to an increase of at least 2 points on the quiz.
Step 2: Choose the significance level (α):
The significance level, also known as the type 1 error rate, is the probability of rejecting the null hypothesis when it is true. In this case, a 5% type 1 error is allowed. Therefore, α = 0.05.
Step 3: Calculate the test statistic:
In this case, since we are testing whether the increase on the quiz score is at least 2 points, we can calculate the difference between the quiz score with the next number of homework problems and the quiz score with the current number of homework problems.
For example, for the first two data points:
Quiz Score: 2 - 1 = 1 (Difference is -1)
Quiz Score: 3 - 2 = 1 (Difference is -1)
Step 4: Determine the critical value or p-value:
To determine the critical value, we need to find the appropriate test statistic distribution. Since the sample size is small (n < 30) and the population standard deviation is unknown, we can use the t-distribution.
With the t-distribution, we need to calculate the degrees of freedom (df = n - 1 = 9 - 1 = 8) and find the critical t-value at the desired significance level (α = 0.05). This critical value will give us the cutoff point for rejecting the null hypothesis.
Alternatively, we can find the p-value associated with the calculated test statistic. The p-value represents the probability of observing a test statistic as extreme as the one obtained, assuming that the null hypothesis is true. If the p-value is less than the significance level (α), we reject the null hypothesis.
Step 5: Make a decision:
If the test statistic is less than the critical value or the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
When performing the calculations, the test statistic will be the mean of the differences between the quiz scores for each level of homework problems.
Using the given data:
Differences: -1, -1, -2, -2, -2, -3, -3, -3, -5
Step 6: Calculate the mean and standard deviation of the differences:
Mean (M): (sum of differences) / (number of differences) = (-1 + -1 + -2 + -2 + -2 + -3 + -3 + -3 + -5) / 9 = -22/9 = -2.44
Standard Deviation (s): calculate the sample standard deviation of the differences
Step 7: Calculate the test statistic:
t = (M - μ₀) / (s / √n)
Where:
M: mean of differences
μ₀: hypothesized mean (0 in this case, since the null hypothesis states that there is no increase)
s: standard deviation of the differences
n: number of differences
Step 8: Determine the critical value or p-value:
Using the t-distribution with (n - 1) degrees of freedom (8 df) and α = 0.05, we can find the critical t-value or calculate the p-value associated with the test statistic.
Step 9: Make a decision:
If the calculated t-value is less than the critical t-value or p-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Note: The actual calculations for the test statistic and critical values or p-value require knowledge of statistical software or tables.