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Chapter 18: Problem 26

Back to campus In 2004 ACT, Inc., reported that \(74 \%\) of 1644 randomly selected college freshmen returned to college the next year. The study was stratified by type of college - public or private. The retention rates were \(71.9 \%\) among 505 students enrolled in public colleges and \(74.9 \%\) among 1139 students enrolled in private colleges. a) Will the \(95 \%\) confidence interval for the true national retention rate in private colleges be wider or narrower than the \(95 \%\) confidence interval for the retention rate in public colleges? Explain. b) Do you expect the margin of error for the overall

Short Answer

Expert verified
Private colleges will have a narrower confidence interval.

Step by step solution

01

Identify Known Values for Public Colleges

Given that 71.9% of 505 students enrolled in public colleges returned the next year. This means the sample proportion \( \hat{p}_{\text{public}} = 0.719 \) and the sample size \( n_{\text{public}} = 505 \).
02

Identify Known Values for Private Colleges

Given that 74.9% of 1139 students enrolled in private colleges returned the next year. This means the sample proportion \( \hat{p}_{\text{private}} = 0.749 \) and the sample size \( n_{\text{private}} = 1139 \).
03

Calculate the Standard Error for Each College Type

The standard error for a proportion is calculated using the formula: \( SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \).- For public colleges: \( SE_{\text{public}} = \sqrt{ \frac{0.719 \times (1 - 0.719)}{505} } \).- For private colleges: \( SE_{\text{private}} = \sqrt{ \frac{0.749 \times (1 - 0.749)}{1139} } \).
04

Calculate the Confidence Interval Width

The width of the confidence interval is roughly proportional to the margin of error, which is \( Z \cdot SE \) (where \( Z \) for a 95% confidence level is approximately 1.96).- For public colleges: Width \( \propto 1.96 \times SE_{\text{public}} \).- For private colleges: Width \( \propto 1.96 \times SE_{\text{private}} \).
05

Compare the Sample Sizes and Their Effects on Widths

Confidence intervals become narrower with larger sample sizes due to the decrease in standard error. The private colleges have a larger sample size (1139) compared to public colleges (505), suggesting the confidence interval for private colleges will be narrower.
06

Conclusion for Part a

The 95% confidence interval for the true national retention rate in private colleges will be narrower than the interval for public colleges because the sample size for private colleges is larger, reducing the standard error and the margin of error.

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Key Concepts

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

Retention Rate
The retention rate in education reflects the percentage of students who return to a college or university in the subsequent year.
This measure is crucial for understanding how well institutions maintain their student body.

In the context of the given problem, the retention rate is differentiated between public and private colleges.
- Public colleges experienced a retention rate of 71.9%.
- Private colleges had a slightly higher retention rate of 74.9%.

These rates can help institutions identify areas for improvement in academic programming and student services.
Retention rates can also influence policy-making and funding allocation.
Sample Proportion
The sample proportion is an important concept in statistics, representing the fraction of observations in a sample that shares a particular attribute.
In simpler terms, it鈥檚 the percentage of a group that displays a specific characteristic.

For public colleges, if 71.9% of the 505 students returned, the sample proportion is calculated as:
\( \hat{p}_{\text{public}} = \frac{505 \times 0.719}{505} = 0.719 \).
Similarly, for private colleges, with a 74.9% retention rate among 1139 students:
\( \hat{p}_{\text{private}} = \frac{1139 \times 0.749}{1139} = 0.749 \).

Understanding the sample proportion helps in estimating population parameters, paving the way for accurate decision-making in educational policies.
Standard Error
Standard error is a measure that indicates the variability or spread of a sample statistic.
In essence, it鈥檚 used to quantify the precision of a sample mean or proportion.

Standard error for a proportion is calculated using:
\[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]
where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.

- For public colleges, this is:
\( SE_{\text{public}} = \sqrt{ \frac{0.719(1-0.719)}{505} } \).
- For private colleges, it鈥檚:
\( SE_{\text{private}} = \sqrt{ \frac{0.749(1-0.749)}{1139} } \).

Smaller standard errors indicate more reliable estimates of population parameters.
They also lead to narrower confidence intervals, providing more precise information about the retention rates.
Margin of Error
The margin of error delineates the range surrounding a sample estimate within which the true population parameter is expected to fall, with a certain level of confidence.
For a 95% confidence interval, the margin of error is calculated as:
\[ ME = Z \cdot SE \]
where \( Z \) is the Z-score corresponding to the confidence level, which is approximately 1.96 for 95%.

- For public colleges, it鈥檚:
1.96 times \( SE_{\text{public}} \).
- For private colleges, it鈥檚:
1.96 times \( SE_{\text{private}} \).

A smaller margin of error equates to a narrower confidence interval, offering a more precise estimate of the retention rate.
Larger sample sizes typically reduce the margin of error, enhancing the reliability of the statistical inferences.

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Most popular questions from this chapter

Another pilot study During routine screening, a doctor notices that \(22 \%\) of her adult patients show higher than normal levels of glucose in their blood-a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within \(4 \%\) with \(98 \%\) confidence. How many randomly selected adults must they test?

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Junk mail Direct mail advertisers send solicitations (a.k.a. "junk mail") to thousands of potential customers in the hope that some will buy the company's product. The acceptance rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people randomly selected from their mailing list of over 200,000 people. They get orders from 123 of the recipients. a) Create a \(90 \%\) confidence interval for the percentage of people the company contacts who may buy something. b) Explain what this interval means. c) Explain what "90\% confidence" means. d) The company must decide whether to now do a mass mailing. The mailing won't be cost-effective unless it produces at least a \(5 \%\) return. What does your confidence interval suggest? Explain.

Pilot study A state's environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of \(3 \%\) and \(90 \%\) confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emissions systems. How many should be sampled for a full investigation?

Mislabeled seafood In December \(2011,\) Consumer Reports published their study of labeling of seafood sold in New York, New Jersey, and Connecticut. They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(22 \%\) of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store or restaurant employees. a) Construct a \(95 \%\) confidence interval for the proportion of all seafood packages in those three states that are mislabeled or misidentified. b) Explain what your confidence interval says about seafood sold in these three states. c) A 2009 report by the Government Accountability Board says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 190 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.
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