Build a mean confidence interval by choosing a t-procedure, checking conditions, calculating endpoints, and interpreting the population parameter.
A population is normally distributed with mean 25. Consider all samples of size 10 . The variable 10sxˉ−25
has a normal distribution.
has a t-distribution with d f=10.
has a t-distribution with d f=9.
has either a normal or a t-distribution depending on the characteristics of the population standard deviation.
C
A company sells a certain type of whistle. The price of the whistle varies from store to store. Julio, a statistician at the company, wants to estimate the mean price, in dollars ($), of this type of whistle at all stores that sell the whistle.
| Scoring for Question 6 Each essentially correct (E) part counts as 1 point, and each partially correct (P) part counts as 21 point. | |
|---|---|
| Score | |
| Complete Response | 4 |
| Substantial Response | 3 |
| Developing Response | 2 |
| Minimal Response | 1 |
| If a response is between two scores (for example, 221 points), use a holistic approach to decide whether to score up or down, depending on the strength of the response and quality of the communication. | |
Identify the appropriate inference procedure for Julio to use.
Model Solution
Use a one-sample t-interval for a population mean.
Scoring components
1. Identifies a one-sample t-interval for a population mean.
Describe the parameter for the inference procedure you identified in part (a-i) in context.
Model Solution
The parameter of interest is the mean price, in dollars, of this type of whistle at all stores that sell the whistle.
Scoring components
2. Identifies the population mean.
3. Includes the population and the whistle price in context.
Consider your work in part (c).
Julio's inference procedure in part (a-i) needs one of the following requirements to be satisfied to verify the normality condition.
- The sample size is greater than or equal to 30.
- If the sample size is less than 30, the distribution of the sample data is not strongly skewed and does not have outliers.
Using your response to (d-i) and the preceding requirements, is the normality condition satisfied for Julio's data? Explain your response.
No, based on the response to part (d-i), Julio's data would not satisfy the normality condition because neither of the criteria listed are met. Julio only has a sample size of 20, which is less than 30, and Pearson's coefficient of skewness indicates the distribution of sample data is strongly skewed.
Essentially correct (E) if the response satisfies the following five components:
In part (d-i) the response indicates the sample is strongly skewed or makes a statement consistent with the graph in part (c)
In part (d-i) the response uses or refers to the graph or the Pearson coefficient of skewness to make the correct determination of skewness
In part (d-ii) the response states that the condition of normality is not met or makes a statement consistent with the explanation in part (d-i)
In part (d-ii) the response explains the sample size of 20 is less than 30
In part (d-ii) the response explains that the distribution of sample data is strongly skewed or makes a statement consistent with the explanation in part (d-i)
Partially correct (P) if the response does not meet the criteria for E but satisfies three or four of components 1-5 required for E.
Incorrect (I) if the response does not meet the criteria for E or P.
Additional Notes:
- A response that determines the sample is strongly skewed, but says the normality condition is satisfied, should be scored I.
- In part (d-i) a response that indicates the distribution is strongly skewed to the left does not satisfy component 1.
- If a response in part (c-ii) has an x on the curve, the response in part (d-i) must indicate the sample is skewed (not strongly skewed or approximately symmetric), or indicate it is not possible to choose whether it is strongly skewed or approximately symmetric to satisfy component 1.
- In part (d-i) a response that indicates the distribution is skewed, but not strongly skewed, because the Pearson coefficient of skewness is not that far from the curve on the graph satisfies component 1 and component 2.
- In part (d-i) a response that indicates the distribution is skewed, but not strongly skewed, may satisfy components 3-5 if in part (d-ii) the response indicates the conditions are met because the distribution is not very skewed and there are no outliers. The response does not need to indicate the sample size is less than 30.
- In part (d-ii) a response that indicates the sample size is small without explicitly referring to the values 20 and 30 may satisfy component 4.
- The quality of communication, or the number of components satisfied for responses with score P, should be considered if holistic scoring is required.