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 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
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

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

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2:14 minutes

Problem 3.CRE.7

Textbook Question

Normal Distribution Examine the distribution shown in the histogram from Exercise 6. Does it appear that the sample data are from a population with a normal distribution? Why or why not?

Verified step by step guidance

Step 1: Observe the histogram provided. A normal distribution typically has a bell-shaped curve, where the frequencies are highest at the center and taper off symmetrically towards the tails. In this histogram, the frequencies do not follow this pattern.

Step 2: Note the peaks and troughs in the histogram. For a normal distribution, the frequencies should gradually decrease as you move away from the center. However, the histogram shows irregular peaks, such as the high frequency at digit 5 and lower frequencies at digits 3 and 4.

Step 3: Assess the symmetry of the distribution. A normal distribution is symmetric around its mean. In this histogram, the frequencies are not symmetric; for example, the frequencies for digits 0, 1, and 2 differ significantly from those for digits 7, 8, and 9.

Step 4: Consider the spread of the data. A normal distribution has a smooth, continuous spread. The histogram here shows a jagged pattern with abrupt changes in frequency, which is inconsistent with a normal distribution.

Step 5: Conclude based on the observations. The irregular shape, lack of symmetry, and uneven spread of frequencies suggest that the sample data are not from a population with a normal distribution.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, where the mean, median, and mode are all equal. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, which is crucial for understanding the spread and central tendency of the data.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. The height of each bar represents the frequency of data points within that interval. Histograms are useful for visualizing the shape of the data distribution, identifying patterns, and detecting outliers, which can help in assessing whether the data follows a normal distribution.

Skewness

Skewness refers to the asymmetry of the distribution of values in a dataset. A distribution is said to be positively skewed if it has a longer tail on the right side, while a negatively skewed distribution has a longer tail on the left. Assessing skewness is important when determining if a dataset approximates a normal distribution, as significant skewness indicates deviations from normality, which can affect statistical analyses.

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Textbook Question

Normal Distribution If the following data are randomly selected, which are expected to have a normal distribution?

a. Weights of Reese’s Peanut Butter Cups

Textbook Question

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.

Between 1.50 and 2.00

Textbook Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

Textbook Question

Normal Distribution If the following data are randomly selected, which are expected to have a normal distribution?

d. Exact volumes of Coke in 12 oz cans

Textbook Question

Standard Normal Distribution. In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Textbook Question

Less than -2.00

Textbook Question

IQ Scores. In Exercises 5–8, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler IQ test).

Textbook Question

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