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 3.3.5c

Chapter 3, Problem 3.3.5c

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.

Verified step by step guidance

Step 1: Recall the formula for calculating a z-score:

z = \frac{X - μ}{σ}

, where

X

is the data value,

μ

is the mean, and

σ

is the standard deviation.

Step 2: Identify the values given in the problem:

X

(highest diastolic blood pressure) = 98 mm Hg,

μ

(mean) = 70.2 mm Hg, and

σ

(standard deviation) = 11.2 mm Hg.

Step 3: Substitute the values into the z-score formula:

z = \frac{98 - 70.2}{11.2}

Step 4: Perform the subtraction in the numerator:

98 - 70.2

Step 5: Divide the result of the subtraction by the standard deviation

11.2

to find the z-score, rounding to two decimal places.

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Key Concepts

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

Z Score

A z score, or standard score, indicates how many standard deviations an element is from the mean of a data set. It is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. Z scores are useful for comparing values from different distributions and understanding their relative position within a distribution.

Mean

The mean, or average, is a measure of central tendency that summarizes a set of values by dividing the sum of all values by the number of values. In the context of the diastolic blood pressure measurements, the mean provides a reference point to understand the typical blood pressure level among the sampled females. It is essential for calculating z scores, as it represents the baseline from which deviations are measured.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. In this exercise, the standard deviation of 11.2 mm Hg helps to understand how individual diastolic blood pressure measurements vary from the mean, which is crucial for calculating z scores.

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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

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

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.

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

Textbook Question

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

Textbook Question

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.