Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 25m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

3. Describing Data Numerically

Standard Deviation

Statistics

3. Describing Data Numerically

Standard Deviation

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1:43 minutes

Problem 3.q.9

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.

Verified step by step guidance

Identify the range of the data by subtracting the minimum value from the maximum value. In this case, the range is calculated as:

R = x_{\max} - x_{\min} = 194 - 10

Recall the range rule of thumb, which states that the standard deviation can be approximated as one-fourth of the range. The formula is:

s = \frac{R}{4}

Substitute the calculated range into the formula for the standard deviation. This gives:

s = \frac{184}{4}

Simplify the fraction to estimate the standard deviation. Perform the division:

s = 184 \div 4

Interpret the result as the estimated standard deviation of the sample of roller coaster maximum speeds, based on the range rule of thumb.

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Key Concepts

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

Range Rule of Thumb

The range rule of thumb is a simple method for estimating the standard deviation of a dataset. It states that the standard deviation can be approximated by dividing the range of the data (the difference between the maximum and minimum values) by four. This rule provides a quick way to gauge variability without complex calculations.

Standard Deviation

Standard deviation is a statistical measure 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 that the values are spread out over a wider range. It is crucial for understanding the distribution of data points in a sample.

Sample Size

Sample size refers to the number of observations or data points collected in a study. In this case, the sample size is 92 roller coaster maximum speeds. A larger sample size generally leads to more reliable estimates of population parameters, such as the standard deviation, as it reduces the impact of outliers and variability in the data.

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Related Practice

Textbook Question

In Exercises 21–28, use the same list of cell phone radiation levels given for Exercises 17–20. Find the indicated percentile or quartile.

Textbook Question

In Exercises 21–28, use the same list of cell phone radiation levels given for Exercises 17–20. Find the indicated percentile or quartile.

P50

Textbook Question

Variance of Roller Coaster Speeds The standard deviation of the sample values in Exercise 1 is 43.1 km/h. What is the variance (including units)?

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

Outliers Identify any of the differences found from Exercise 1 that appear to be outliers. For any outliers, how much of an effect do they have on the mean, median, and standard deviation?

Textbook Question

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 29–32, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed using all of the data.

Body Temperatures Refer to Data Set 5 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2.

Textbook Question

Finding Standard Deviation from a Frequency Distribution. In Exercises 37–40, refer to the frequency distribution in the given exercise and compute the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 18.5 minutes; (Exercise 38) 36.7 minutes; (Exercise 39) 6.9 years; (Exercise 40) 20.4 seconds.

Standard deviation for frequency distribution

Textbook Question

The Empirical Rule Based on Data Set 1 “Body Data” in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/) Using the empirical rule, what is the approximate percentage of women with platelet counts

a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

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