1. Mrs. Chauver has an interesting approach to assigning grades in her statistios elass. Of the 5 tests sfudents take throughout the semester, Mrs, Chativet selects a fandomi sample of 3 , finds the average score of these tests, and records this average as the student's final grade. Joe's test scores are as follows, \( 93,87,96,78,90 \), (a) List all 10 possible samples of size 3, \( 93,87,96 \) \( 93,78,90 \) \( 87,93,96 \) \( 87,78,90 \) (b) Calculate the mean of each sample and display the sampling distribution of the sample mean using a dotplot.

Certainly! Let's work through the problem step by step. ## Problem 1 Mrs. Chauver assigns final grades by taking a random sample of 3 out of 5 test scores and averaging them. Joe's test scores are as follows: \[ 93, \ 87, \ 96, \ 78, \ 90 \] ### Part (a): List All 10 Possible Samples of Size 3 To determine all possible samples of size 3 from Joe's 5 test scores, we can use combinations since the order of selection does not matter. The number of combinations is given by: \[ C(5, 3) = \frac{5!}{3!(5-3)!} = 10 \] Here are all **10 possible samples**: 1. **93, 87, 96** 2. **93, 87, 78** 3. **93, 87, 90** 4. **93, 96, 78** 5. **93, 96, 90** 6. **93, 78, 90** 7. **87, 96, 78** 8. **87, 96, 90** 9. **87, 78, 90** 10. **96, 78, 90** ### Part (b): Calculate the Mean of Each Sample and Display the Sampling Distribution Let's calculate the mean for each of the 10 samples: 1. **93, 87, 96** \[ \text{Mean} = \frac{93 + 87 + 96}{3} = \frac{276}{3} = 92 \] 2. **93, 87, 78** \[ \text{Mean} = \frac{93 + 87 + 78}{3} = \frac{258}{3} = 86 \] 3. **93, 87, 90** \[ \text{Mean} = \frac{93 + 87 + 90}{3} = \frac{270}{3} = 90 \] 4. **93, 96, 78** \[ \text{Mean} = \frac{93 + 96 + 78}{3} = \frac{267}{3} = 89 \] 5. **93, 96, 90** \[ \text{Mean} = \frac{93 + 96 + 90}{3} = \frac{279}{3} = 93 \] 6. **93, 78, 90** \[ \text{Mean} = \frac{93 + 78 + 90}{3} = \frac{261}{3} = 87 \] 7. **87, 96, 78** \[ \text{Mean} = \frac{87 + 96 + 78}{3} = \frac{261}{3} = 87 \] 8. **87, 96, 90** \[ \text{Mean} = \frac{87 + 96 + 90}{3} = \frac{273}{3} = 91 \] 9. **87, 78, 90** \[ \text{Mean} = \frac{87 + 78 + 90}{3} = \frac{255}{3} = 85 \] 10. **96, 78, 90** \[ \text{Mean} = \frac{96 + 78 + 90}{3} = \frac{264}{3} = 88 \] #### Sampling Distribution of the Sample Mean The **sampling distribution of the sample mean** consists of all the sample means calculated above. Here's a summary of the means and their frequencies: | **Sample Mean** | **Frequency** | |-----------------|---------------| | 85 | 1 | | 86 | 1 | | 87 | 2 | | 88 | 1 | | 89 | 1 | | 90 | 1 | | 91 | 1 | | 92 | 1 | | 93 | 1 | ##### Dotplot Representation While I can't provide a visual dotplot here, I can describe how you would create one: 1. **Draw a horizontal number line** that spans from the lowest mean (85) to the highest mean (93). 2. **Plot each sample mean** as a dot above its corresponding value on the number line. 3. **Stack dots vertically** if multiple samples have the same mean. For example: ``` 85 | • 86 | • 87 | •• 88 | • 89 | • 90 | • 91 | • 92 | • 93 | • ``` This dotplot shows the distribution of Joe's sample means, with **85** and **93** appearing once, **86** and **88-92** also appearing once each, and **87** appearing twice. #### Interpretation - **Range of Sample Means**: The means range from 85 to 93. - **Most Frequent Mean**: 87 appears twice, making it the most frequent sample mean. - **Symmetry**: The distribution is fairly symmetric around the center with a slight concentration towards the middle values. This sampling distribution helps Mrs. Chauver understand the variability in Joe's final grade based on different samples of his test scores.