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Ch. 3 - Describing, Exploring, and Comparing Data
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 3 - Describing, Exploring, and Comparing DataProblem 3.2.45d
Chapter 3, Problem 3.2.45d

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.)


d. Which approach results in values that are better estimates of part (b) or part (c)? Why? When computing variances of samples, should you use division by n or

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1
Step 1: Understand the problem. The question is asking about the difference between dividing by n (the sample size) versus dividing by n-1 when calculating sample variance. This is a fundamental concept in statistics related to unbiased estimation.
Step 2: Recall the formulas for variance. For a population variance, the formula is: σ2=∑(x-μ)2n. For a sample variance, the formula is: s2=∑(x-x̄)2n-1. The key difference is the denominator: n for population variance and n-1 for sample variance.
Step 3: Understand why we divide by n-1 for sample variance. Dividing by n-1 instead of n corrects for bias in the estimation of the population variance. This adjustment is known as Bessel's correction. It ensures that the sample variance is an unbiased estimator of the population variance.
Step 4: Relate this to the problem. When computing variances of samples, using division by n-1 provides better estimates of the population variance because it accounts for the fact that the sample mean is itself an estimate and introduces variability. Dividing by n would underestimate the population variance.
Step 5: Answer the question about which approach results in better estimates. Using division by n-1 (sample variance formula) results in values that are better estimates of the population variance. This is because it adjusts for the bias introduced by using the sample mean as an estimate of the population mean.

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Key Concepts

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

Population and Sample

In statistics, a population refers to the entire group of individuals or instances about which we seek to draw conclusions, while a sample is a subset of that population selected for analysis. Understanding the distinction is crucial because statistical methods often rely on samples to estimate population parameters, such as means and variances, especially when it is impractical to collect data from the entire population.
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Sampling Distribution of Sample Proportion

Variance and Sample Variance

Variance is a measure of how much values in a dataset differ from the mean of that dataset. When calculating the variance of a sample, we use the formula that divides by n-1 (where n is the sample size) instead of n to account for the fact that we are estimating the population variance from a sample. This adjustment, known as Bessel's correction, helps to provide an unbiased estimate of the population variance.
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Sampling with Replacement

Sampling with replacement means that after an item is selected from the population, it is returned before the next selection, allowing the same item to be chosen multiple times. This method affects the independence of samples and the calculations of probabilities and variances, as it maintains the same population size for each selection, which can lead to different statistical properties compared to sampling without replacement.
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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

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.


d. Using the criteria summarized in Figure 3-6, is the commute time of 95 minutes significantly low, significantly high, or neither?

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

Percentile Use the weights from Exercise 1 to find the percentile for 3647 mg.

Textbook Question

Roller Coaster z Score A larger sample of 92 roller coaster maximum speeds has a mean of 85.9 km/h and a standard deviation of 28.7 km/h. What is the z score for a speed of 34 km/h? Does the z score suggest that the speed of 34 km/h is significantly low?

Textbook Question

Estimating s The sample of 92 roller coaster maximum speeds includes values ranging from a low of 10 km/h to a high of 194 km/h. Use the range rule of thumb to estimate the standard deviation.