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

2. Describing Data with Tables and Graphs

Histograms

Statistics

2. Describing Data with Tables and Graphs

Histograms

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2:18 minutes

Problem 2.2.2

Textbook Question

More IQ Scores The population of IQ scores of adults is normally distributed. If we obtain a voluntary response sample of 5000 of those IQ scores, will a histogram of the sample be bell-shaped?

Verified step by step guidance

Understand the concept of a voluntary response sample: This type of sample is self-selected, meaning individuals choose to participate. It often leads to bias because those who respond may not represent the entire population.

Recall the properties of a normal distribution: A normal distribution is symmetric and bell-shaped, characterized by its mean and standard deviation.

Consider the Central Limit Theorem: This theorem states that the distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the population distribution. However, it applies to the distribution of sample means, not the distribution of individual sample values.

Evaluate the impact of sample bias: Since the sample is voluntary, it may not accurately reflect the population's normal distribution. This could affect the shape of the histogram, making it potentially not bell-shaped.

Conclude based on sample characteristics: Due to the voluntary nature of the sample, the histogram may not be bell-shaped even if the population distribution is normal. The sample may be skewed or have other biases that affect its shape.

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Key Concepts

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

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean and standard deviation, where most of the data points cluster around the mean, and the probabilities for values taper off equally in both directions. Understanding this concept is crucial as it describes the population of IQ scores in the question.

Voluntary Response Sample

A voluntary response sample is a type of non-probability sampling where participants self-select into the sample. This method often leads to bias because individuals with strong opinions or interests are more likely to participate, which may not accurately represent the entire population. Recognizing this bias is essential to evaluate whether the sample's histogram will reflect the population's normal distribution.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where data is grouped into bins or intervals. The height of each bar reflects the frequency of data points within each interval. In the context of the question, understanding how a histogram can visually represent the distribution of IQ scores helps assess whether the sample's distribution is bell-shaped like the population's normal distribution.

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Related Practice

Textbook Question

Modified Box-and-Whisker Plot In Exercises 59–62, (a) identify any outliers and (b) draw a modified box-and-whisker plot that represents the data set. Use asterisks (*) to identify outliers.

75 78 80 75 62 72 74 75 80 95 76 72

Multiple Choice

Use the frequency histogram below to determine (a) the number of classes and (b) the class width.

Textbook Question

In Exercises 9–18, construct the histograms and answer the given questions.

Burger King Dinner Service Times Use the frequency distribution from Exercise 18 in Section 2-1 to construct a histogram. Using a strict interpretation of the criteria for being a normal distribution, does the histogram appear to depict data from a population with a normal distribution?

Textbook Question

IQ Scores IQ scores of adults are normally distributed. If a large sample of adults is randomly selected and the IQ scores are illustrated in a histogram, what is the shape of that histogram?

Textbook Question

Cell Phone Radiation If we collect a sample of cell phone radiation amounts much larger than the sample included with Exercise 3, and if our sample includes a single outlier, how will that outlier appear in a histogram?

Textbook Question

In Exercises 5–8, answer the questions by referring to the following Minitab-generated histogram, which depicts the weights (grams) of all quarters listed in Data Set 40 “Coin Weights” in Appendix B. (Grams are actually units of mass and the values shown on the horizontal scale are rounded.)

Sample Size What is the approximate number of quarters depicted in the three bars farthest to the left?

Textbook Question

Class Width and Class Limits Give the approximate values of the class width, and the lower and upper class limits of the class depicted in the bar farthest to the left.

Textbook Question

Relative Frequency Histogram How would the shape of the histogram change if the vertical scale uses relative frequencies expressed in percentages instead of the actual frequency counts as shown here?

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