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Chapter 9: Problem 37

Appropriate use of the interval $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. a. \(n=50\) and \(\hat{p}=0.30\) b. \(n=50\) and \(\hat{p}=0.05\) c. \(n=15\) and \(\hat{p}=0.45\) d. \(n=100\) and \(\hat{p}=0.01\)

Short Answer

Expert verified
The sample size is large enough for the cases where \(n=50\) and \(\hat{p}=0.30\), and \(n=15\) and \(\hat{p}=0.45\). The sample sizes are not large enough for the cases where \(n=50\) and \(\hat{p}=0.05\), and \(n=100\) and \(\hat{p}=0.01\).

Step by step solution

01

For n=50 and \(\hat{p}\)=0.30

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(50*0.30 = 15\) and \(50*(1-0.30) = 35\). Because both of these values exceed 5, the sample size in this scenario is large enough.
02

For n=50 and \(\hat{p}\)=0.05

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(50*0.05 = 2.5\) and \(50*(1-0.05)=47.5\). Since the value of \(n\hat{p}\) is less than 5, the sample size in this scenario is not large enough.
03

For n=15 and \(\hat{p}\)=0.45

Calculate values of \(n\hat{p}\) and \(n(1-\hat{p})\). In this case, \(15*0.45=6.75\) and \(15*(1-0.45)=8.25\). Both of these are greater than 5, indicating that the sample size in this scenario is large enough.
04

For n=100 and \(\hat{p}\)=0.01

Calculate values of \(n\hat{p}\) and \(n*(1-\hat{p})\). Here, the calculations yield \(100*0.01=1\) and \(100*(1-0.01)=99\). The value of \(n\hat{p}\) is less than 5, which indicates that the sample is not large enough in this case.

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Key Concepts

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

Sample Size Determination
Sample size determination is a crucial step in statistical analysis, especially when crafting confidence intervals. Confidence intervals help us estimate population parameters, and an adequate sample size ensures that these estimates are reliable. To determine if the sample size is large enough for a binomial proportion, we use the rule of thumb: both \( n\hat{p} \) and \( n(1-\hat{p}) \) must be greater than or equal to 5. This ensures the sampling distribution of the proportion is approximately normal.
  • Calculate \( n\hat{p} \) by multiplying the sample size \( n \) with the estimated proportion \( \hat{p} \).
  • Calculate \( n(1-\hat{p}) \) to check the size of the complementary proportion.
If both calculations exceed 5, the sample size is deemed sufficient for estimating the confidence interval appropriately. This criterion helps to ensure that there is enough data to support meaningful statistical conclusions.
Binomial Proportion
Understanding binomial proportion is essential in statistics, especially when you are working with categorical data. A binomial proportion refers to the ratio of the number of successes to the total number of trials in a binomial experiment. In the context of confidence intervals, \( \hat{p} \), the sample proportion, represents an estimate of this ratio.
  • The binomial proportion \( \hat{p} \) is calculated by dividing the number of successes by the sample size \( n \).
  • This proportion is crucial for constructing the confidence interval, as it provides an estimate of the population proportion \( p \).
In practice, \( \hat{p} \) is used not only in calculating confidence intervals but also in hypothesis testing and other statistical analyses. It represents how often a particular outcome occurs in your sample, making it a foundational concept in statistics.
Statistical Assumptions
In statistical analyses, making the right assumptions is vital for the validity of your results. When creating confidence intervals for binomial proportions, certain assumptions need to be met to ensure accuracy and precision.
  • **Independence**: The trials must be independent, meaning the outcome of one trial does not affect another.
  • **Fixed number of trials**: There should be a fixed number of observations or trials \( n \).
  • **Binary outcomes**: Each trial must result in a binary outcome, such as success or failure.
  • **Large sample size**: As discussed earlier, \( n\hat{p} \) and \( n(1-\hat{p}) \) should both be at least 5 to validate the normal approximation.
These assumptions are paramount to apply the confidence interval formula correctly. If any of these assumptions are violated, the results from the confidence interval may not be reliable, and alternate statistical methods might be necessary.

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Most popular questions from this chapter

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who are registered to vote. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) I. \(\quad n=1,000, p=0.5\) II. \(\quad n=200, p=0.6\) III. \(\quad n=100, p=0.7\)

In response to budget cuts, county officials are interested in learning about the proportion of county residents who favor closure of a county park rather than closure of a county library. In a random sample of 500 county residents, 198 favored closure of a county park. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0429 Statement 2: The estimate \(\hat{p}=0.396\) will never differ from the value of the actual population proportion by more than 0.0429 Statement 3: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0219

a. Use the given information to estimate the proportion of college students who use the Internet more than 3 hours per day. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.Most American college students make use of the Internet for both academic and social purposes. What proportion of students use it for more than 3 hours a day? The authors of the paper "U.S. College Students" Internet Use: Race, Gender and Digital Divides" (Journal of Computer-Mediated Communication [2009]: 244-264) describe a survey of 7,421 students at 40 colleges and universities. The sample was selected to reflect general demographics of U.S. college students. Of the students surveyed, 2,998 reported Internet use of more than 3 hours per day.

For estimating a population characteristic, why is an unbiased statistic with a small standard error preferred over an unbiased statistic with a larger standard error?

For each of the following choices, explain which would result in a narrower large-sample confidence interval for \(p\) : a. \(95 \%\) confidence level or \(99 \%\) confidence level b. \(n=200\) or \(n=500\)
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