Twenty students randomly assigned to an experimental group receive an instructional program: 30 in a control group do not. After 6 months, both groups are tested on their knowledge.
The experimental group has a mean of 38 on the test(with an estimated population standard deviation of 3): the control group has a mean of 35 ( with an estimated population standard deviation of 5) Using the .05 level, what should the experimenter conclude?
(a)Use the steps of hypothesis testing,
(b) sketch the distributions involved
(c) explain your answer to someone who is familiar with the t test for a single sample, but not with the t test for independent means.
(a) and (c)
Ho: u(ex) - u(ct) = 0
Ha:u(ex) – u(ct) is not equal to 0
Test stat: t = (38-35) / sqrt[9/20 + 25/30] = 2.6482
p-value = 2*P(t > 2.6482 with df = 48) = 0.0109
Conclusion:
At the 5% significance level, reject Ho because the p-value is less than 5%
I need part b any suggestions?
(a) The steps of hypothesis testing can be followed to determine what the experimenter should conclude. Here are the steps to perform the hypothesis testing:
Step 1: State the null and alternative hypotheses
- Null hypothesis (H₀): The instructional program has no effect on the knowledge test scores. The mean score of the experimental group (µ₁) is equal to the mean score of the control group (µ₂).
- Alternative hypothesis (H₁): The instructional program has an effect on the knowledge test scores. The mean score of the experimental group (µ₁) is significantly different from the mean score of the control group (µ₂).
Step 2: Set the significance level
- The significance level, denoted by α, is defined as the probability of rejecting the null hypothesis when it is true. In this case, α is given as 0.05.
Step 3: Compute the test statistic
- Since the population standard deviations are estimated, we need to use the t-test for independent means. The formula for the t-test for independent means is:
t = (x₁ - x₂) / sqrt((s₁²/n₁) + (s₂²/n₂))
Where:
- x₁ and x₂ are the sample means of the experimental and control groups, respectively.
- s₁ and s₂ are the estimated population standard deviations of the experimental and control groups, respectively.
- n₁ and n₂ are the sample sizes of the experimental and control groups, respectively.
Step 4: Determine the critical value
- The critical value is the value beyond which we reject the null hypothesis. We need to refer to the t-distribution table or use a statistical software to find the critical value corresponding to the chosen significance level and degrees of freedom.
Step 5: Make a decision
- If the absolute value of the calculated test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
(b) To sketch the distributions involved, we need to plot two normal distributions representing the test scores of the experimental and control groups. The mean of the experimental group's distribution is 38, with an estimated population standard deviation of 3. The mean of the control group's distribution is 35, with an estimated population standard deviation of 5. The distributions should be labeled accordingly.
(c) I will now explain the answer to someone familiar with the t-test for a single sample but not with the t-test for independent means.
In this experiment, we have two groups of students: the experimental group that received an instructional program and the control group that did not. We want to determine if the instructional program had a significant effect on the test scores after 6 months.
To do that, we can use the t-test for independent means, which compares the means of two independent groups. This test allows us to see if there is a significant difference between the mean scores of the two groups.
In our case, the null hypothesis states that there is no difference in the mean test scores of the experimental and control groups. The alternative hypothesis states that there is a significant difference in the mean test scores of the two groups.
We calculate the test statistic, which measures how far the sample means are from each other, taking into account the estimated population standard deviations and the sample sizes of each group. Using the t-distribution, we determine the critical value that separates the values we would expect to see if the null hypothesis is true from the values that would suggest the alternative hypothesis is true.
Finally, we compare the calculated test statistic to the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in test scores between the two groups. If the test statistic is not greater than the critical value, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in test scores.