Let's work step-by-step through the problem to find the needed information.
1. Identify the standard deviations of both samples:
- Standard deviation of the red box, [tex]\( \sigma_1 \)[/tex], is 3.868
- Standard deviation of the blue box, [tex]\( \sigma_2 \)[/tex], is 2.933
2. Identify the sample sizes:
- The sample size for the red box, [tex]\( N_1 \)[/tex], is 100
- The sample size for the blue box, [tex]\( N_2 \)[/tex], is 100
3. Calculate the standard deviation of the sample mean differences (let's denote it as [tex]\( \sigma_{\Delta \bar{X}} \)[/tex]):
The formula to calculate the standard deviation of the sample mean differences when two sample sizes and their standard deviations are given is:
[tex]\[ \sigma_{\Delta \bar{X}} = \sqrt{ \left( \frac{\sigma_1^2}{N_1} \right) + \left( \frac{\sigma_2^2}{N_2} \right)} \][/tex]
Substituting the given values:
[tex]\[ \sigma_{\Delta \bar{X}} = \sqrt{ \left( \frac{3.868^2}{100} \right) + \left( \frac{2.933^2}{100} \right)} \][/tex]
From the given data, after applying the necessary calculations, the result we obtain is:
[tex]\[ \sigma_{\Delta \bar{X}} \approx 0.4854 \][/tex]
Now, we summarize the findings in the given statements:
- The sample size of the session regarding the number of people who would purchase the red box, [tex]\( N_1 \)[/tex], is [tex]\( 100 \)[/tex].
- The sample size of the session regarding the number of people who would purchase the blue box, [tex]\( N_2 \)[/tex], is [tex]\( 100 \)[/tex].
- The standard deviation of the sample mean differences is approximately [tex]\( 0.485 \)[/tex].
