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Chapter 7: Problem 31

Find the sample size required to estimate the population mean. Assume that all grade-point averages are to be standardized on a scale between 0 and \(4 .\) How many grade-point averages must be obtained so that the sample mean is within 0.01 of the population mean? Assume that a \(95 \%\) confidence level is desired. If we use the range rule of thumb, we can estimate \(\sigma\) to be range/4 \(=(4-0) / 4=1 .\) Does the sample size seem practical?

Short Answer

Expert verified
38416 grade-point averages are required for the desired accuracy at 95% confidence level.

Step by step solution

01

- Identify Variables

Note the given information: - Confidence level: 95%, which gives a Z-value of 1.96 - Standard deviation, \(\sigma\), estimated as 1 (from the range rule of thumb) - Margin of error, \(\text{E}\text{, is 0.01}\)
02

- Use the Sample Size Formula

The formula for calculating sample size is: \[n = \left(\frac{Z \sigma}{E}\right)^2\]Substitute the values: - \(\text{Z} = 1.96\) - \(\sigma = 1\) - \(\text{E} = 0.01\)
03

- Plug in the Values

Substitute and compute: \[n = \left(\frac{1.96 \times 1}{0.01}\right)^2\] \[n = \left(196\right)^2\] \[n = 38416\text{..} \]
04

- Interpret the Result

The sample size required is 38,416. To achieve a mean within 0.01 of the population mean with a 95% confidence level, a very large sample size is needed.

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Key Concepts

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

Population Mean
The population mean is the average value of a set of measurements or values of the entire population. In our problem, it represents the average grade-point average (GPA) of all students. It is denoted by the symbol \(\mu\).

Understanding the population mean is critical when drawing conclusions from sample data. It gives us a measure to compare individual scores and sample statistics. In real-world applications, we often don't have access to the entire population, so we estimate this mean using a sample mean \(\bar{x}\).
Confidence Level
The confidence level indicates the probability that the value of a parameter falls within a specified range of values. In our example, a 95% confidence level means we are 95% certain that the sample mean will be within the margin of error from the population mean.

Confidence levels are represented typically as percentages (like 90%, 95%, and 99%). They are linked to Z-values in the context of the normal distribution. For a 95% confidence level, the corresponding Z-value is approximately 1.96. This Z-value is what we use in our sample size formula to ensure that our estimate has the desired confidence.
Margin of Error
The margin of error is the maximum amount by which the sample mean is expected to differ from the actual population mean. In our exercise, we desired a margin of error of 0.01 for the GPA. This means we want our sample mean to be within 0.01 points of the actual population mean.

A smaller margin of error requires a larger sample size because we want to be more precise in our estimation. The margin of error is represented by \(E\) and directly influences the sample size calculation:

= \left( \frac{Z \sigma}{E} \right) ^2
Standard Deviation
Standard deviation, denoted by \(\sigma\), measures the amount of variability or dispersion of a set of values. In our case, it represents the standard deviation of GPAs. A higher standard deviation means the values are more spread out.

Using the range rule of thumb, we can estimate the standard deviation as the range divided by four. For GPAs that range from 0 to 4, the standard deviation is: \[ \sigma = \frac{4-0}{4} = 1 \].

In our sample size calculation, the standard deviation helps determine how spread out our data are and affects how large our sample size needs to be to achieve the desired margin of error and confidence level.

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Most popular questions from this chapter

Use the data and confidence level to construct a confidence interval estimate of \(p,\) then address the given question. When she was 9 years of age. Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in "A Close Look at Therapeutic Touch," Journal of the American Medical Association, Vol. 279, No. 13). a. Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses? b. Using Emily's sample results, what is the best point estimate of the therapists success rate? c. Using Emily's sample results, construct a \(99 \%\) confidence interval estimate of the proportion of correct responses made by touch therapists.

Use the data and confidence level to construct a confidence interval estimate of \(p,\) then address the given question. The drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.). a. Construct a \(95 \%\) confidence interval estimate of the percentage of OxyContin users who develop nausea. b. Compare the result from part (a) to this \(95 \%\) confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: \(1.93 \%

Use the given data to find the minimum sample size required to estimate a population proportion or percentage. You plan to develop a new iOS social gaming app that you believe will surpass the success of Angry Birds and Facebook combined. In forecasting revenue, you need to estimate the percentage of all smartphone and tablet devices that use the iOS operating system versus Android and other operating systems. How many smartphones and tablets must be surveyed in order to be \(99 \%\) confident that your estimate is in error by no more than two percentage points? a. Assume that nothing is known about the percentage of portable devices using the iOS operating system. b. Assume that a recent survey suggests that about \(43 \%\) of smartphone and tablets are using the iOS operating system (based on data from NetMarkesShare). c. Does the additional survey information from part (b) have much of an effect on the sample size that is required?

Finding Confidence Intervals. In Exercises \(9-16,\) assume that each sample is a simple random sample obtained from a population with a normal distribution. Garlic for Reducing Cholesterol In a test of the effectiveness of garlic for lowering cholesterol, 49 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment. The changes (before minus after) in their levels of LDL cholesterol (in \(\mathrm{mg} / \mathrm{dL}\) ) had a mean of 0.4 and a standard deviation of 21.0 (based on data from "Effect of Raw Garlic vs Commercial Garlic Supplements on Plasma Lipid Concentrations in Adults with Moderate Hypercholesterolemia," by Gardner et al., Archives of Internal Medicine, Vol. 167). Construct a \(98 \%\) confidence interval estimate of the standard deviation of the changes in LDL cholesterol after the garlic treatment. Does the result indicate whether the treatment is effective?

Use the given data to find the minimum sample size required to estimate a population proportion or percentage. In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. Find the sample size needed to estimate that percentage. Use a 0.05 margin of error, and use a confidence level of \(95 \% .\) a. Assume that nothing is known about the percentage to be estimated. b. Assume that prior studies have shown that about \(40 \%\) of full-time students earn bachelor's degrees in four years or less. c. Does the added knowledge in part (b) have much of an effect on the sample size?
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