12 Jul Sample Mean Distribution And T-Interval
Title: Sample Mean Distribution and T-Interval
Task 1: Read the following case study, titled “Green M&M’s”:
Green M&M’s: Consider a class of 20 statistics students, where each student is given 5 small bags of M&M’s and asked to count the number of green M&M’s. The results are shown below. The population mean is for a bag of this size is 10.06 and the population standard deviation is 2.59.
| Bag 1 | Bag 2 | Bag 3 | Bag 4 | Bag 5 | |
| Student 1 | 3 | 4 | 7 | 6 | 14 |
| Student 2 | 12 | 14 | 18 | 8 | 3 |
| Student 3 | 12 | 18 | 8 | 13 | 11 |
| Student 4 | 18 | 12 | 7 | 11 | 8 |
| Student 5 | 17 | 18 | 9 | 14 | 2 |
| Student 6 | 3 | 10 | 14 | 9 | 13 |
| Student 7 | 3 | 12 | 11 | 9 | 15 |
| Student 8 | 6 | 2 | 3 | 18 | 11 |
| Student 9 | 8 | 16 | 12 | 17 | 3 |
| Student 10 | 14 | 13 | 11 | 17 | 5 |
| Student 11 | 3 | 14 | 17 | 17 | 15 |
| Student 12 | 7 | 14 | 11 | 7 | 2 |
| Student 13 | 17 | 2 | 12 | 18 | 13 |
| Student 14 | 9 | 18 | 8 | 11 | 10 |
| Student 15 | 14 | 16 | 4 | 3 | 12 |
| Student 16 | 4 | 3 | 7 | 11 | 14 |
| Student 17 | 11 | 17 | 6 | 5 | 13 |
| Student 18 | 15 | 8 | 17 | 11 | 10 |
| Student 19 | 4 | 9 | 13 | 16 | 16 |
| Student 20 | 12 | 12 | 5 | 14 | 16 |
Answer the following questions:
- Find the sample means for each student’s green M&M count.
- Create a histogram of the sample means, and calculate the mean and standard deviation of the sample means. To construct a histogram, follow the given steps:
- I. Obtain a frequency (relative-frequency, percent) distribution of the data.
- II. Draw a horizontal axis on which to place the bars and a vertical axis on which to display the frequencies (relative frequencies, percents).
- III. For each class, construct a vertical bar whose height equals the frequency (relative frequency, percent) of that class.
- IV. Label the bars with the classes, the horizontal axis with the name of the variable, and the vertical axis with “Frequency” (“Relative frequency,” “Percent”).
Source: Weiss, Neil A. (2012). Elementary Statistics (8th ed.). Upper Saddle River, NJ: Pearson.
- Theoretically, what are the mean, standard deviation, and distribution of all possible sample means for a sample size of 5?
Task 2: Read the following case study, titled “Diamond Pricing”:
In a Singapore Edition of Business Times, diamond pricing was explored. The price of a diamond is based on the diamond’s weight, color, and clarity. A simple random sample of 18 one-half-carat diamonds had the following prices, in dollars:
| 1676 | 1442 | 1995 | 1718 | 1826 | 2071 | 1947 | 1983 | 2146 |
| 1995 | 1876 | 2032 | 1988 | 2071 | 2234 | 2108 | 1941 | 2316 |
Based on the above information, solve the following problems:
- Apply the t-interval procedure to these data to find a 90% confidence interval for the mean price of all one-half-carat diamonds. Interpret your result. (Note: ) Obtain a normal probability plot, a boxplot, a histogram, and a stem-and-leaf diagram of the data.
- b. Based on your graphs from part (b), is it reasonable to apply the t-interval procedure as you did in part a? Explain your answer.
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