# Statistics for Social Science

Statistics for Social Science takes a fresh approach to the introductory class. With learning check questions, embedded videos and interactive simulations, students engage in active learning as they read. An emphasis on real-world and academic applications help ground the concepts presented. Designed for students taking an introductory statistics course in psychology, sociology or any other social science discipline.

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## Comparison of Social Sciences Textbooks

Consider adding Top Hat’s Statistics for Social Sciences textbook to your upcoming course. We’ve put together a textbook comparison to make it easy for you in your upcoming evaluation.

### Top Hat

Steve Hayward et al., Statistics for Social Sciences, Only one edition needed

### Pearson

Agresti, Statistical Methods for the Social Sciences, 5th Edition

### Cengage

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

### Sage

Gregory Privitera, Essentials Statistics for the Behavioral Sciences, 2nd Edition

### Pricing

Average price of textbook across most common format

#### $200.83 Hardcover print text only ####$239.95

Hardcover print text only

#### $92 Hardcover print text only ### Always up-to-date content, constantly revised by community of professors Content meets standard for Introduction to Anatomy & Physiology course, and is updated with the latest content ### In-Book Interactivity Includes embedded multi-media files and integrated software to enhance visual presentation of concepts directly in textbook Only available with supplementary resources at additional cost Only available with supplementary resources at additional cost Only available with supplementary resources at additional cost ### Customizable Ability to revise, adjust and adapt content to meet needs of course and instructor ### All-in-one Platform Access to additional questions, test banks, and slides available within one platform ## Pricing Average price of textbook across most common format ### Top Hat Steve Hayward et al., Statistics for Social Sciences, Only one edition needed #### Up to40-60%more affordable Lifetime access on any device ### Pearson Agresti, Statistical Methods for the Social Sciences, 5th Edition ####$200.83

Hardcover print text only

### Pearson

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

#### $239.95 Hardcover print text only ### Sage McConnell, Brue, Flynn, Principles of Microeconomics, 7th Edition ####$92

Hardcover print text only

## Always up-to-date content, constantly revised by community of professors

Constantly revised and updated by a community of professors with the latest content

### Top Hat

Steve Hayward et al., Statistics for Social Sciences, Only one edition needed

### Pearson

Agresti, Statistical Methods for the Social Sciences, 5th Edition

### Pearson

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

### Sage

Gregory Privitera, Essentials Statistics for the Behavioral Sciences, 2nd Edition

## In-book Interactivity

Includes embedded multi-media files and integrated software to enhance visual presentation of concepts directly in textbook

### Top Hat

Steve Hayward et al., Statistics for Social Sciences, Only one edition needed

### Pearson

Agresti, Statistical Methods for the Social Sciences, 5th Edition

### Pearson

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

### Sage

Gregory Privitera, Essentials Statistics for the Behavioral Sciences, 2nd Edition

## Customizable

Ability to revise, adjust and adapt content to meet needs of course and instructor

### Top Hat

Steve Hayward et al., Statistics for Social Sciences, Only one edition needed

### Pearson

Agresti, Statistical Methods for the Social Sciences, 5th Edition

### Pearson

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

### Sage

Gregory Privitera, Essentials Statistics for the Behavioral Sciences, 2nd Edition

## All-in-one Platform

### Top Hat

Steve Hayward et al., Statistics for Social Sciences, Only one edition needed

### Pearson

Agresti, Statistical Methods for the Social Sciences, 5th Edition

### Pearson

Gravetter et al., Essentials of Statistics for The Behavioral Sciences, 9th Edition

### Sage

Gregory Privitera, Essentials Statistics for the Behavioral Sciences, 2nd Edition

A lifelong learner, Steve focused on statistics and research methodology during his graduate training at the University of New Mexico. He later founded and served as CEO of Center for Performance Technology, providing instructional design and training development support to larger client organizations throughout the United States. Steve is presently lead faculty member for statistics at Rio Salado College in Tempe, Arizona.

#### Joseph F. Crivello, PhDUniversity of Connecticut

Joseph Crivello has taught Anatomy & Physiology for over 34 years, and is currently a Teaching Fellow and Premedical Advisor of the HMMI/Hemsley Summer Teaching Institute.

## Explore this textbook

Read the fully unlocked textbook below, and if you’re interested in learning more, get in touch to see how you can use this textbook in your course today.

# Confidence Intervals

## Chapter Objectives

After completing this chapter, you will be able to:

• Describe the role of estimation within inferential statistics
• Distinguish between confidence levels and confidence intervals
• Calculate and interpret confidence intervals for means and proportions
• Apply abstract concepts like sampling error to specific methods like surveys and polls
• Find and interpret the margin of error and apply it to a confidence interval
• Describe the impact of changing the sample size on our ability to make estimates

## What Are Confidence Intervals and Why Do We Use Them?

By this point in your reading, it has likely dawned on you that statistics have a great deal of applicability to everyday life. Statistics involve everything from the calculation of your Grade Point Average to deciding whether you should invest in the stock market. Day to day life revolves around statistical probability. It is human nature to want to predict what might occur tomorrow, next month, next year, and so forth, as it gives us a sense of control over what happens to us.

A confidence interval (C.I.) is a type of probability statistic that specifies the range of values we use to estimate the location of an unknown population parameter such as a mean or a proportion. In other words, we don’t usually know the population mean or proportion so we estimate the likelihood of it to be within a certain range on the basis of a sample mean or proportion. The width of that range is determined by something called the margin of error (E). The margin of error accompanies a confidence level, which tells us how likely our parameter is to lie within that interval. Together, they tell us how accurate our estimate is likely to be. Because sample statistics are used to estimate the location of population parameters, confidence intervals lie within the realm of inferential statistics.

You might have encountered confidence intervals quite a bit without even realizing it. Suppose that you have been feeling fatigued and you visit your physician for a blood sample to see if you are anemic. The method to determine whether your red blood cell count is in the “normal” range involves the use of a confidence interval. Or if you have read in the newspaper about the latest Gallup poll on Americans’ attitudes toward gun control, you will likely see a statement about the margin of error that is part of a confidence interval.

Be it in medicine, political polling, or many other areas of our everyday life, confidence intervals are calculated in a variety of situations in which we need to gauge the reliability of our estimates. Some of these situations involve critical decision-making (such as whether your blood test is far enough outside of the “normal” range to warrant medical treatment), while others provide us with knowledge for its own sake. Below we will examine how social researchers use confidence intervals in different situations to estimate the probability that the parameter in which they are interested lies within a particular range of scores.

## Point Estimates and Interval Estimates

A point estimate is a single statistic (usually a sample mean or a percentage) that is used as our best estimate of a corresponding parameter, i.e., the value in a population from which you drew that estimate. For example, a sample of people who voted in the last Canadian election (such as the Canadian Elections Survey) might give us a mean age of 50.2 years, which we could then use as an estimate of the mean age of the Canadian population who voted.

Point estimates are appealingly simple. They possess a lot of apparent precision in that they give us a specific figure (in this case one that suggests youth electoral participation is very low). But it would be somewhat misleading because we cannot be certain that the true population mean is exactly 50.2 years. The point estimate’s level of precision comes at the expense of its accuracy. Accuracy does not refer to mistakes per se, but to freedom from error, or the extent to which our estimate differs from the parameter of interest’s true value (in this case the true age of voters in the population). Data that we get from samples will always possess some degree of error.

For this reason, researchers tend to calculate an interval estimate, which provides us with a range of values within which the population parameter is likely to fall. For example, between 46.2 years and 54.2 years of age. We therefore would have a less precise (but more accurate) idea of what the average age of Canadian voters will likely be, give or take a little bit. Confidence intervals, as we shall see, are a type of interval estimate.

### Examples: Point Estimates vs. Interval Estimates

Example 1: Estimate of a Mean

The amount of time that children spend in front of a screen (including activities like texting, television, and video games) has been of increasing interest to social researchers as technology moves more deeply into our daily lives. Suppose that the mean amount of daily screen time recorded for a sample of 500 children was 4.0 hours. Some children might have as little as one hour or less of screen time, while others might spend over 5.0 hours in front of one, but our best estimate for an average child (drawn from the entire population of children across the country) is precisely 4.0 hours.

We could instead estimate that the mean amount of screen time spent by children likely falls somewhere between 2.5 and 5.5 hours. While it would be less precise than saying it is exactly 4.0 hours, it is probably more accurate to use a range of values. That would be a fairly wide estimate, and it is important not to make the estimate so wide that it becomes of little utility (wherein you are pretty much guaranteed that the parameter will fall within the interval simply because it includes nearly all possible values).

Example 2: Estimate of a Proportion (in Percent)

Polling research shows that there are certain topics that are particularly likely to divide populations along political and ideological lines. One of those topics is the death penalty. Trend studies have shown that the percentage of Americans who support the death penalty has gradually decreased since the 1970s, but remains above 50%. In 2015, Gallup measured support for the death penalty for murder at 61% of all Americans (although much lower for Blacks and Hispanics). We cannot know for sure how many Americans support it, but our best estimate is 61% (expressed as a proportion would be .61).

If we were to state that “between 57% and 65% of Americans support the death penalty for murder”, we would be making an interval estimate. It is not as precise as saying “61%,” but it is likely to be more accurate in that we can have greater confidence that the true percentage is somewhere in that range.

8.01

Which of the following would be an example of a point estimate? (Select all that apply)

A

The percentage of Gallup poll respondents who said that they intended to vote in the next U.S. presidential election.

B

The percentage of all registered voters in the United States who are female is between 45 and 55 percent.

C

The mean age of a sample of registered voters from the most recent U.S. presidential election.

D

The average temperature of newborn infants is within three degrees of 96.8º F.

8.02

Imagine that you were studying the weight of children in the United Kingdom. Which of the following would be an example (or examples) of an interval estimate? (Select all that apply)

A

The mean weight of British 5-year-olds is 40 pounds.

B

The proportion of children who are overweight is estimated to be between 35% to 45%.

C

40% of all 5-year-old children in Great Britain are overweight.

D

The average weight of all British 5-year-olds is between 39.5 and 44.5 pounds.

8.03

While the main advantage of a point estimate is its _____, its main disadvantage is its _____.

A

Precision; inaccuracy

B

Accuracy; imprecision

## The Level of Confidence

The essence of sampling theory is that a well-drawn probability sample (wherein every element has a known probability of being selected, normally involving a random mechanism) will usually yield results that very closely resemble those that we would get if we measured the entire population. Researchers use samples because measuring entire populations is usually not only unfeasible but also unnecessary. The law of large numbers tells us that with a sufficiently large sample size, sample statistics (usually means) will tend to approximate population parameters very closely. You will find some values that are higher than expected, and some that are lower, but in a large sample those differences tend to average out.

However, we do take a certain amount of risk with using any sample to make inferences about populations. Because they are based on samples, interval estimates, including confidence intervals, will vary from sample to sample, as illustrated in Figure 8.3 below. The number of those lines representing the means of different samples (x̅1, x̅2, etc.) could theoretically carry on infinitely. Most of them will contain the population parameter, while a certain proportion will not. A confidence level specifies the probability that our particular sample’s interval estimate will in fact contain the population parameter.

The researcher typically sets the desired confidence level at the outset, depending upon how much risk he or she is willing to take. The lower the confidence level, the more risk that their interval estimate does not contain the parameter. The higher the confidence level, the lower that risk. The law of large numbers tells us that the larger the sample size, the lower the risk that our sample will not contain the parameter; therefore, we can select a higher level of confidence with larger sample sizes.

## The Three Most Common Levels of Confidence

Conventionally there are three levels of confidence used in the social sciences: 90%, 95%, and 99%. Most often, academic researchers and pollsters using large samples (at least 30 cases, although opinions on that number vary) prefer the 95% or the 99% confidence level.

What this means is that in, for example, 95% of all samples drawn from the same population using the same measure (e.g. a polling question) at the same time will be equal to ± the amount of sampling error in the confidence interval in question. In the case of the 99% confidence level, 99% of all samples drawn from the same population using the same measure at the same time will also be ± the amount of sampling error in question.

In other words, how confident are we that our results are accurate? What is the chance that our intervals do not contain the true population value?

## Common Misinterpretations of Confidence Level

Many statisticians are at pains to remind non-statisticians that confidence levels are very easily misunderstood. One common misinterpretation is that your interval estimate will hold a certain percentage of all cases in the population. For example, one could mistakenly conclude that 95% of all of the weights of adult males in the United States will be between 150 and 165 pounds. A correct interpretation would be: We can be 95% confident that our interval estimate of 150 to 165 pounds will in fact contain the true population value.

A 90% confidence level does not mean for example that 90% of all sample means in a sampling distribution will fall within our confidence interval. It means that in 10% (or 1 in 10) of all samples (if we were theoretically to repeat our sample over and over, infinitely), we would expect to fail to capture the population mean within our interval estimate. Likewise, a 95% confidence level means that we would expect to fail to capture the population mean in 5% (or 5 in 100) of our samples, or 1% (1 in 100) of our samples at the 99% confidence level.

Examples: Levels of Confidence

### Class Discussion 8.04

Class Discussion 8.04

When should we not calculate a confidence interval?

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## Answers to Case Study Questions

### Answer to Case Study Question 8.01

The religious composition of the public population coincides with the declining prevalence of traditional religious beliefs and practices.

### Answer to Case Study Question 8.02

Age. As older generations or cohorts of adults pass away, younger adults with far less levels of attachment to a religious affiliation and its practices replace them.

### Answer to Case Study Question 8.03

According to the 2014 U.S. Religious Landscape Study, with a nationally representative sample of 35,071, nine-in-ten adults claim religious institutions strengthen community bonds, and three quarters claim that these institutions protect and strengthen morality in overall society.

## Answers to Pre-Class Discussion Questions

### Answer to Class Discussion 8.01

The population is usually too large to measure.

### Answer to Class Discussion 8.02

When you have a large sample or a known population standard deviation. It is rare to have a known population standard deviation but a common scenario is IQ, since it has been measured in so many studies. Sampling theory shows us that larger samples have less sampling error and we can safely assume that the sample standard deviation approximates the population standard deviation.

### Answer to Class Discussion 8.03

Your friend would win the bet. The trade-off for choosing a higher confidence level is that there is a wider MOE.