Practise describing quantitative distributions through shape, centre, variability, and unusual features, then use graphical or numerical evidence to justify claims.
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.
Julio wants to examine some characteristics of the distribution of the sample of whistle prices.
Julio called the managers of 20 randomly selected stores that sell the whistle and recorded the price of the whistle at each store. Following is a dotplot of Julio's data.
Price of the Whistle at 20 Stores
The summary statistics for Julio's data are shown in the following table.
| Sample Size | Mean | Standard Deviation | Minimum | Median | Maximum | ||
|---|---|---|---|---|---|---|---|
| 20 | 5.12 | 0.743 | 4.25 | 4.51 | 4.885 | 5.475 | 6.58 |
Summary Statistics for Julio's Data
Describe the shape of the distribution of the sample of whistle prices. Justify your response using appropriate values from the summary statistics table.
It can often be difficult to determine whether the distribution of sample data is skewed by looking at a graph of the data and the summary statistics, particularly when the sample size is small. Thus, statisticians sometimes measure how skewed a data set is. One such measure is Pearson's coefficient of skewness, which is calculated using the following formula.
In the formula, is the sample mean, m is the sample median, and s is the sample standard deviation.
Calculate Pearson's coefficient of skewness for Julio's sample of 20 whistle prices. Show your work.
The following graph shows conclusions that can be made about the shape of the distribution of sample data based on Pearson's coefficient of skewness and sample size.
Conclusion from Pearson's Coefficient of Skewness
Indicate the value of the Pearson's coefficient of skewness you calculated in part (c-i) for the appropriate sample size by marking it with an "X" on the preceding graph.
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.
What should you conclude about the shape of the distribution of the sample of whistle prices? Justify your response.
The following histograms summarize the teaching year for the teachers at two high schools, A and B.
High School A
High School B
Teaching year is recorded as an integer, with first-year teachers recorded as 1, second-year teachers recorded as 2, and so on. Both sets of data have a mean teaching year of 8.2, with data recorded from 200 teachers at High School A and 221 teachers at High School B. On the histograms, each interval represents possible integer values from the left endpoint up to but not including the right endpoint.
The standard deviation of the teaching year for the 221 teachers at High School B is 7.2. If one teacher is selected at random from High School B, what is the probability that the teaching year for the selected teacher will be within 1 standard deviation of the mean of 8.2 ? Justify your answer.