Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

Statistics

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

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1:50 minutes

Problem 3.3.5c

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.

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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Related Practice

Textbook Question

Distributions In a continuous uniform distribution,

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a. Find the mean and standard deviation for the distribution of the waiting times represented in Figure 6-2, which accompanies Exercises 5–8.

Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

e. Convert the longest wait time to a z score.

f. Based on the result from part (e), is the longest wait time significantly high?

Textbook Question

Pulse Rates. In Exercises 13–24, use the data in the table below for pulse rates of adult males and females (based on Data Set 1 “Body Data” in Appendix B). Hint: Draw a graph in each case.

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Find the probability that a male has a pulse rate between 70 beats per minute and 90 beats per minute.

Textbook Question

Pulse Rates. In Exercises 13–24, use the data in the table below for pulse rates of adult males and females (based on Data Set 1 “Body Data” in Appendix B). Hint: Draw a graph in each case.

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For males, find P90 which is the pulse rate separating the bottom 90% from the top 10%.

Multiple Choice

Sketch a graph to represent the probability, then use a calculator to find it.

$P\left(Z>1.14\right)$

Multiple Choice

Use a calculator to find the z–scores of the region shown in the standard normal distribution below.

A = 0.800

Textbook Question

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Textbook Question

Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

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