Twenty students assigned to an experimental group receive 30 in a control 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 distributors involved
(c)explain your answer to someone who is familar with the test for a single sample, but not independent means

(a) The steps of hypothesis testing:

1. State the null hypothesis (H0) and the alternative hypothesis (Ha).
2. Determine the significance level (α).
3. Collect the data and calculate the test statistic.
4. Find the critical value(s) or p-value.
5. Make a decision and interpret the results.
6. State the conclusion.

In this case, the null hypothesis (H0) is that there is no difference between the means of the experimental group and the control group. The alternative hypothesis (Ha) is that there is a significant difference between the means.

The significance level is given as 0.05.

To perform the hypothesis test, we need to calculate the test statistic. Since we are comparing means of two independent groups, we can use the independent samples t-test.

The formula for the test statistic is:
t = (mean1 - mean2) / √((s1^2 / n1) + (s2^2 / n2))

Where:
mean1 = mean of the experimental group
mean2 = mean of the control group
s1 = estimated population standard deviation of the experimental group
s2 = estimated population standard deviation of the control group
n1 = sample size of the experimental group
n2 = sample size of the control group

Substituting the given values:
mean1 = 38
mean2 = 35
s1 = 3
s2 = 5
n1 = 20
n2 = 30

t = (38 - 35) / √((3^2 / 20) + (5^2 / 30))

Next, we need to find the critical value or p-value.

Since our alternative hypothesis is that there is a significant difference, it is a two-tailed test. We can find the critical value using the t-distribution table or a statistical software. At α = 0.05 level of significance, with degrees of freedom (df) = n1 + n2 - 2 = 20 + 30 - 2 = 48, the critical value is approximately ±2.011.

If the calculated test statistic t falls outside the critical value range, we reject the null hypothesis. If the calculated test statistic t falls within the critical value range, we fail to reject the null hypothesis.

Finally, we state our conclusion based on the decision made in step 5.

(b) Sketching the distributions involved:
The distribution for the experimental group and control group can be sketched using normal distribution curves. The experimental group distribution would have a mean of 38 and a standard deviation of 3, while the control group distribution would have a mean of 35 and a standard deviation of 5. The distributions will overlap, and we can compare the means to determine the difference.

(c) Explanation for someone familiar with the test for a single sample, but not independent means:
In this scenario, we have two groups - an experimental group and a control group. We want to compare the means of these two groups to determine if there is a significant difference. The test we can use is called an independent samples t-test. It is similar to the t-test for a single sample, but instead of comparing the sample mean to a population mean, we compare the means of two independent groups. We calculate a test statistic that takes into account the means, standard deviations, and sample sizes of both groups. We then compare it to the critical value at a given significance level (α). If the test statistic falls within the critical value range, we fail to reject the null hypothesis. If it falls outside the critical value range, we reject the null hypothesis and conclude that there is a significant difference between the means of the two groups.

To determine what the experimenter should conclude, we need to perform hypothesis testing for independent means. Here are the steps involved in hypothesis testing:

Step 1: State the Null Hypothesis (H0) and Alternative Hypothesis (H1)
The null hypothesis assumes that there is no significant difference between the means of the experimental and control group, while the alternative hypothesis assumes that there is a significant difference between the means.

H0: μ1 = μ2 (There is no significant difference between the means of the experimental and control group)
H1: μ1 ≠ μ2 (There is a significant difference between the means of the experimental and control group)

Step 2: Set the Level of Significance (α)
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is true. In this case, the level of significance is given as .05, which means we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true).

Step 3: Calculate the Test Statistic
The test statistic we will use for this scenario is the independent samples t-test. This test statistic measures the difference between the means of two independent groups, taking into account the sample sizes and variances.

Step 4: Determine the Critical Region
The critical region is the range of test statistic values that, if our calculated test statistic falls within, would lead us to reject the null hypothesis. The critical region is determined by the level of significance (α) and the degrees of freedom (df).

Step 5: Calculate the P-value
The P-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the one observed, assuming that the null hypothesis is true. We can use the P-value to determine whether to reject or fail to reject the null hypothesis. If the P-value is less than the level of significance (α), we reject the null hypothesis.

Step 6: Make a Conclusion
Based on the critical region or the P-value, we make a decision whether to reject or fail to reject the null hypothesis. If the test statistic falls within the critical region or the P-value is less than α, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Now, to explain this answer to someone familiar with the test for a single sample but not independent means:

The experimenter is trying to determine if there is a significant difference in means between the experimental and control groups. The null hypothesis assumes that there is no significant difference between the means, while the alternative hypothesis assumes that there is a significant difference. The level of significance is set at .05, which means if the probability of obtaining the observed test statistic or one more extreme under the null hypothesis is less than .05, we will reject the null hypothesis. We calculate the test statistic using the independent samples t-test, which measures the difference between means while considering sample sizes and variances. By comparing the test statistic to the critical region determined by the level of significance and degrees of freedom, or by calculating the P-value, we can make a decision whether to reject or fail to reject the null hypothesis.