Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.2.45b

Chapter 3, Problem 3.2.45b

Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)
b. After listing the nine different possible samples of two values selected with replacement, find the sample variance (which includes division by ) for each of them; then find the mean of the nine sample variances s2.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the sample variance for all possible samples of size 2 (with replacement) from the population {9, 10, 20}. Then, we calculate the mean of these sample variances. Sample variance is calculated using the formula: s² = (Σ(xᵢ - x̄)²) / (n - 1), where x̄ is the sample mean, xᵢ are the sample values, and n is the sample size.

Step 2: List all possible samples of size 2 with replacement. Since the population has 3 values (9, 10, 20), and sampling is done with replacement, the total number of samples is 3 × 3 = 9. The samples are: (9, 9), (9, 10), (9, 20), (10, 9), (10, 10), (10, 20), (20, 9), (20, 10), (20, 20).

Step 3: For each sample, calculate the sample mean (x̄). For example, for the sample (9, 9), the mean is x̄ = (9 + 9) / 2 = 9. Repeat this calculation for all 9 samples.

Step 4: For each sample, calculate the sample variance (s²). Use the formula s² = (Σ(xᵢ - x̄)²) / (n - 1), where n = 2. For example, for the sample (9, 9), the variance is s² = [(9 - 9)² + (9 - 9)²] / (2 - 1) = 0. Repeat this calculation for all 9 samples.

Step 5: Find the mean of the 9 sample variances. Add up all the sample variances calculated in Step 4 and divide by 9 (the total number of samples). This gives the mean of the sample variances.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Variance

Sample variance is a measure of how much the values in a sample differ from the sample mean. It is calculated by taking the sum of the squared differences between each sample value and the sample mean, then dividing by the number of observations minus one (n-1). This adjustment (using n-1) corrects for bias in the estimation of the population variance from a sample.

Random Sampling with Replacement

Random sampling with replacement means that each selected value is returned to the population before the next selection. This method ensures that each selection is independent and that the probability of selecting any value remains constant throughout the sampling process. It is crucial for calculating probabilities and variances in statistical analysis.

Mean of Sample Variances

The mean of sample variances is the average of the variances calculated from multiple samples. It provides an overall estimate of the variability within the population based on the sampled data. This mean is particularly useful in understanding the distribution of variances and can help in making inferences about the population variance.

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Related Practice

Textbook Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

Diastolic Blood Pressure of Females For the diastolic blood pressure measurements of females listed in Data Set 1 “Body Data” in Appendix B, the highest measurement is 98 mm Hg. The 147 diastolic blood pressure measurements of females have a mean of 70.2 mm Hg and a standard deviation of 11.2 mm Hg.

c. Convert the highest diastolic blood pressure to a z score.

Textbook Question

Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n); then find the mean of those nine population variances.

Textbook Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

New York City Commute Time New York City commute times (minutes) are listed in Data Set 31 “Commute Times” in Appendix B. The 1000 times have a mean of 42.6 minutes and a standard deviation of 26.2 minutes. Consider the commute time of 95.0 minutes.

c. Convert the commute time of 95.0 minutes to a z score.

Textbook Question

Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47.

a. Find the missing value.

Textbook Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

b. How many standard deviations is that [the difference found in part (a)]?

Textbook Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

b. How many standard deviations is that [the difference found in part (a)]?