Questions: Homework 6 - Chapter 6 Question 8 of 14 (1 point) I Question Attempt: 1 of Unlimited Consider choosing a marble like sampling from a population. (The population mean of the number of push-ups is μ=7.67 and the population standard deviation is σ=0.47.) (a) Suppose a sample of size 2 is randomly selected from the population, with replacement, as follows. One marble is randomly chosen, the number of push-ups is completed, and the marble is put back into the bag. Then for a second time a marble is randomly chosen and the number of push-ups is completed. There are 9 possible samples. The numbers of push-ups for several of the possible samples have been listed in the table below. Enter the numbers of push-ups for the remaining possible samples. When you are done, select "Compute". In the "Sample mean, x̄ " column, you will then see the sample mean of the numbers of push-ups for each sample, along with the mean and standard deviation of all the column's values. Index Sample Numbers of push-ups Sample mean, x̄ 1 blue, blue 7,7 2 blue, red 7,8 3 blue, green 7,8 4 red, blue 8,7 5 red, red 8,8 6 red, green 8,8 7 green, blue , 8 green, red ,

Homework 6 - Chapter 6 Question 8 of 14 (1 point) I Question Attempt: 1 of Unlimited Consider choosing a marble like sampling from a population. (The population mean of the number of push-ups is μ=7.67 and the population standard deviation is σ=0.47.) (a) Suppose a sample of size 2 is randomly selected from the population, with replacement, as follows. One marble is randomly chosen, the number of push-ups is completed, and the marble is put back into the bag. Then for a second time a marble is randomly chosen and the number of push-ups is completed. There are 9 possible samples. The numbers of push-ups for several of the possible samples have been listed in the table below. Enter the numbers of push-ups for the remaining possible samples. When you are done, select "Compute". In the "Sample mean, x̄ " column, you will then see the sample mean of the numbers of push-ups for each sample, along with the mean and standard deviation of all the column's values. Index Sample Numbers of push-ups Sample mean, x̄ 1 blue, blue 7,7 2 blue, red 7,8 3 blue, green 7,8 4 red, blue 8,7 5 red, red 8,8 6 red, green 8,8 7 green, blue , 8 green, red ,
Transcript text: Homework 6 - Chapter 6 Question 8 of 14 (1 point) I Question Attempt: 1 of Unlimited Consider choosing a marble like sampling from a population. (The population mean of the number of push-ups is $\mu=7.67$ and the population standard deviation is $\sigma=0.47$.) (a) Suppose a sample of size 2 is randomly selected from the population, with replacement, as follows. One marble is randomly chosen, the number of push-ups is completed, and the marble is put back into the bag. Then for a second time a marble is randomly chosen and the number of push-ups is completed. There are 9 possible samples. The numbers of push-ups for several of the possible samples have been listed in the table below. Enter the numbers of push-ups for the remaining possible samples. When you are done, select "Compute". In the "Sample mean, $\bar{x}$ " column, you will then see the sample mean of the numbers of push-ups for each sample, along with the mean and standard deviation of all the column's values. \begin{tabular}{|c|c|c|c|} \hline Index & Sample & \begin{tabular}{c} Numbers \\ of push- \\ ups \end{tabular} & Sample mean, $\overline{\boldsymbol{x}}$ \\ \hline 1 & blue, blue & 7,7 & \\ \hline 2 & blue, red & 7,8 & \\ \hline 3 & blue, green & 7,8 & \\ \hline 4 & red, blue & 8,7 & \\ \hline 5 & red, red & 8,8 & \\ \hline 6 & red, green & 8,8 & \\ \hline 7 & green, blue & $\square, \square$ & \\ \hline 8 & green, red & $\square, \square$ & \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Calculate Sample Means

For each sample of push-ups, we calculate the sample mean \( \bar{x} \) as follows:

  1. Sample 1: \( (7, 7) \)
    \[ \bar{x} = \frac{7 + 7}{2} = 7.0 \]

  2. Sample 2: \( (7, 8) \)
    \[ \bar{x} = \frac{7 + 8}{2} = 7.5 \]

  3. Sample 3: \( (7, 8) \)
    \[ \bar{x} = \frac{7 + 8}{2} = 7.5 \]

  4. Sample 4: \( (8, 7) \)
    \[ \bar{x} = \frac{8 + 7}{2} = 7.5 \]

  5. Sample 5: \( (8, 8) \)
    \[ \bar{x} = \frac{8 + 8}{2} = 8.0 \]

  6. Sample 6: \( (8, 8) \)
    \[ \bar{x} = \frac{8 + 8}{2} = 8.0 \]

  7. Sample 7: \( (8, 7) \)
    \[ \bar{x} = \frac{8 + 7}{2} = 7.5 \]

  8. Sample 8: \( (8, 8) \)
    \[ \bar{x} = \frac{8 + 8}{2} = 8.0 \]

  9. Sample 9: \( (8, 8) \)
    \[ \bar{x} = \frac{8 + 8}{2} = 8.0 \]

Step 2: Calculate Mean of Sample Means

The mean of the sample means \( \mu \) is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N \bar{x}_i}{N} = \frac{69.0}{9} = 7.67 \]

Step 3: Calculate Variance and Standard Deviation

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = 0.11 \]

The standard deviation \( \sigma \) is then:

\[ \sigma = \sqrt{0.11} \approx 0.33 \]

Final Answer

The mean of the sample means is \( \mu = 7.67 \) and the standard deviation of the sample means is \( \sigma \approx 0.33 \).

\[ \boxed{\mu = 7.67, \sigma \approx 0.33} \]

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