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Chapter 9: Problem 6

Construct the appropriate confidence interval. A simple random sample of size \(n=785\) adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a \(95 \%\) confidence interval about the population proportion of adults who follow college football.

Short Answer

Expert verified
The \(95\%\) confidence interval is \(0.3161 \text{ to } 0.3839\).

Step by step solution

01

- Identify the sample statistics

Determine the sample proportion (\(\hat{p}\)). In this case, \(n = 785\), and the number of successes (adults who follow college football) is \(x = 275\). The sample proportion is calculated as follows: \(\hat{p} = \frac{x}{n} = \frac{275}{785} \approx 0.350\).
02

- Find the critical value

For a \(95\%\) confidence interval, the critical value (\(z\)-score) corresponding to the middle \(95\%\) under the standard normal curve is \(z = 1.96\).
03

- Calculate the standard error of the proportion

The standard error (SE) of the sample proportion is given by the formula: \(\text{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Plug in the values: \(\text{SE} = \sqrt{\frac{0.350 \times 0.650}{785}} \approx 0.0173\).
04

- Calculate the margin of error

The margin of error (ME) is found by multiplying the critical value by the standard error: \(\text{ME} = z \times \text{SE} = 1.96 \times 0.0173 \approx 0.0339\).
05

- Construct the confidence interval

The confidence interval for the population proportion is given by \(\hat{p} \pm \text{ME}\). So, \(\text{CI} = 0.350 \pm 0.0339 = (0.3161, 0.3839)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sample proportion
The sample proportion (\(\text{\textbackslash hat\textcurly p}\)) is a statistic that estimates the true proportion of a population based on a random sample. In our given exercise, we have a sample of 785 adults, and we want to know how many of them follow college football. Out of these, 275 adults said they do follow college football. To find the sample proportion, we use the formula: \(\text{\textbackslash hat\textcurly p} = \text{\textbackslash frac\textbackslash (x\textbackslash )\textbackslash (n\textbackslash )}\), where \(\text{\textbackslash frac\text{275}\text{785}}\text-backslash \backslashapprox 0.350\).This value (0.350) tells us that 35% of our sample follow college football. It鈥檚 a simple yet powerful way to represent proportions. New lines make it easier to read.
critical value
The critical value is a factor used to compute the margin of error in a confidence interval. For a confidence interval of 95%, the critical value comes from the standard normal distribution and is denoted as \(\text{\textbackslash z}\). In our example, we use a z-score of 1.96. The critical value depends on the desired confidence level, which determines the

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Most popular questions from this chapter

How much do Americans sleep each night? Based on a random sample of 1120 Americans 15 years of age or older, the mean amount of sleep per night is 8.17 hours according to the American Time Use Survey conducted by the Bureau of Labor Statistics. Assuming the population standard deviation for amount of sleep per night is 1.2 hours, construct and interpret a \(95 \%\) confidence interval for the mean amount of sleep per night of Americans 15 years of age or older.

Construct a confidence interval of the population proportion at the given level of confidence. $$x=400, n=1200,95 \% \text { confidence }$$

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Construct the appropriate confidence interval. In a random sample of 40 felons convicted of aggravated assault, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct and interpret a \(95 \%\) confidence interval for the mean length of sentence for an aggravated assault conviction. (Source: Based on data obtained from the U.S. Department of Justice.)
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