To address the problem, we are given the standard deviations for two different samples, specifically:
- The standard deviation for the red box sample is 3.868
- The standard deviation for the blue box sample is 2.933
We need to find the following values:
1. The sample size of the session regarding the number of people who would purchase the red box ([tex]\(N_1\)[/tex]).
2. The sample size of the session regarding the number of people who would purchase the blue box ([tex]\(N_2\)[/tex]).
3. The standard deviation of the sample mean differences.
Given:
- The sample size for the session regarding the number of people who would purchase the red box, [tex]\(N_1\)[/tex], is [tex]\(30\)[/tex].
- The sample size for the session regarding the number of people who would purchase the blue box, [tex]\(N_2\)[/tex], is [tex]\(30\)[/tex].
Next, we use these sample sizes to determine the standard deviation of the sample mean differences. The formula for the standard deviation of the sample mean differences when dealing with two independent samples is given by:
[tex]\[
\text{Standard Deviation of the Sample Mean Differences} = \sqrt{\frac{\sigma_1^2}{N_1} + \frac{\sigma_2^2}{N_2}}
\][/tex]
where:
- [tex]\(\sigma_1\)[/tex] is the standard deviation of the red box sample.
- [tex]\(\sigma_2\)[/tex] is the standard deviation of the blue box sample.
- [tex]\(N_1\)[/tex] is the sample size of the red box session.
- [tex]\(N_2\)[/tex] is the sample size of the blue box session.
By substituting the given values:
[tex]\[
\sigma_1 = 3.868 \quad \text{and} \quad \sigma_2 = 2.933
\][/tex]
[tex]\[
N_1 = 30 \quad \text{and} \quad N_2 = 30
\][/tex]
The calculation results in:
[tex]\[
\text{Standard Deviation of the Sample Mean Differences} \approx 0.886
\][/tex]
Thus, completing the statements:
1. The sample size of the session regarding the number of people who would purchase the red box, [tex]\(N_1\)[/tex], is [tex]\(30\)[/tex].
2. The sample size of the session regarding the number of people who would purchase the blue box, [tex]\(N_2\)[/tex], is [tex]\(30\)[/tex].
3. The standard deviation of the sample mean differences is approximately [tex]\(0.886\)[/tex].
