Question

course activity: treatments and significance
sample mean differences and standard deviation
as a member of the marketing team for a pasta manufacturer, you want to find out whether theres any difference in the mean number of people who would buy the new macaroni product, l - bow roni, if it had a red box and if it had a blue box.
in each session, you bring in 30 people to try l - bow roni and have them respond with whether they would buy this product over the competitors product. suppose you conducted 45 sessions with the red box and 60 sessions with the blue box. this data gives you the number of yes responses to the survey for each session. note that the two samples are different sizes.
to determine whether this difference is significant, you need to find the standard deviation of the sample mean differences. for that task, youll use this formula for the standard deviation of sample mean differences:
\\(\sigma_{m_1 - m_2}=\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}\\)
in this formula, the subscripts \\(m_1\\) and \\(m_2\\) represent the means of the two samples, \\(\sigma_1\\) and \\(\sigma_2\\) are the standard deviations of the two populations, and \\(n_1\\) and \\(n_2\\) are the sample sizes.
part a
question
use the spreadsheets average function (look under autosum in the function menu) to calculate the means.
enter the correct value in each box. use numerals instead of words, and round each value to the nearest hundredth.
the mean value of people who would purchase the red box is
the mean value of people who would purchase the blue box is
part b
question
what is the difference of the sample means of those who would purchase the red box and those who would purchase the blue box?
1.74
2.08
1.85
1.39
part c
question
use the standard deviation values of the two samples to find the standard deviation of the sample mean differences.
then complete each statement.
the sample size of the session regarding the number of people would purchase the red box, \\(n_1\\), is
the sample size of the session regarding the number of people would purchase the blue box, \\(n_2\\), is
the standard deviation of the sample mean differences is approximately

Explanation:

Step1: Calculate mean for red - box

Since we are not given the actual data values for the red - box sessions to use the spreadsheet's Average function, assume we had the data. If we had \(n_1 = 45\) sessions and the sum of the number of yes - responses for each session is \(S_1\), the mean \(\bar{x}_1=\frac{S_1}{45}\). But without data, we can't calculate the exact value.

Step2: Calculate mean for blue - box

Similarly, for \(n_2 = 60\) sessions and sum of yes - responses \(S_2\), the mean \(\bar{x}_2=\frac{S_2}{60}\).

Step3: Calculate difference of sample means (Part B)

Let the mean of red - box be \(\bar{x}_1\) and blue - box be \(\bar{x}_2\). The difference \(\bar{x}_1-\bar{x}_2\). Since we don't have the means from Step1 and Step2 calculated, we assume if we had \(\bar{x}_1\) and \(\bar{x}_2\) we would subtract them.

Step4: Calculate standard deviation of sample mean differences (Part C)

Let \(\sigma_1\) and \(\sigma_2\) be the standard deviations of the two samples. Given \(n_1 = 45\) and \(n_2=60\), the formula for the standard deviation of sample mean differences is \(\sigma_{\bar{x}_1-\bar{x}_2}=\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}\). We are given \(\sigma_1\) and \(\sigma_2\) values in the table (not shown fully here), we would substitute \(n_1 = 45\), \(n_2 = 60\), \(\sigma_1\) and \(\sigma_2\) into the formula.

Answer:

Part A: Without data, unable to provide values.
Part B: Without calculated means from Part A, unable to determine.
Part C: The sample size of the session regarding the number of people would purchase the red box, \(n_1\) is 45. The sample size of the session regarding the number of people would purchase the blue box, \(n_2\) is 60. Without \(\sigma_1\) and \(\sigma_2\) values fully shown, unable to calculate the standard deviation of the sample - mean differences.