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Chapter 9: Problem 37

Appropriate use of the interval $$ \hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ 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. $$ \begin{array}{l} \text { a. } n=50 \text { and } \hat{p}=0.30 \\ \text { b. } n=50 \text { and } \hat{p}=0.05 \\ \text { c. } n=15 \text { and } \hat{p}=0.45 \end{array} $$ d. \(n=100\) and \(\hat{p}=0.01\)

Short Answer

Expert verified
a. Yes, the sample size is large enough for this interval to be appropriate. b. No, the sample size is not large enough for this interval to be appropriate. c. No, the sample size is not large enough for this interval to be appropriate. d. No, the sample size is not large enough for this interval to be appropriate.

Step by step solution

01

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

Calculate \(n\hat{p}\) and \(n(1 - \hat{p})\): \( n\hat{p} = 50 \cdot 0.30 = 15 \geq 10 \) \( n(1 - \hat{p}) = 50 \cdot (1-0.30) = 50 \cdot 0.70 = 35 \geq 10\) Both values are greater than 10, so the sample size is large enough for this interval to be appropriate.
02

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

Calculate \(n\hat{p}\) and \(n(1 - \hat{p})\): \( n\hat{p} = 50 \cdot 0.05 = 2.5 \) \( n(1 - \hat{p}) = 50 \cdot (1-0.05) = 50 \cdot 0.95 = 47.5 \geq 10\) The value of \(n\hat{p}\) is less than 10, so the sample size is not large enough for this interval to be appropriate.
03

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

Calculate \(n\hat{p}\) and \(n(1 - \hat{p})\): \( n\hat{p} = 15 \cdot 0.45 = 6.75 \) \( n(1 - \hat{p}) = 15 \cdot (1-0.45) = 15 \cdot 0.55 = 8.25 \) Both values are less than 10, so the sample size is not large enough for this interval to be appropriate.
04

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

Calculate \(n\hat{p}\) and \(n(1 - \hat{p})\): \( n\hat{p} = 100 \cdot 0.01 = 1 \) \( n(1 - \hat{p}) = 100 \cdot (1-0.01) = 100 \cdot 0.99 = 99 \geq 10\) The value of \(n\hat{p}\) is less than 10, so the sample size is not large enough for this interval to be appropriate.

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Key Concepts

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

Sample Size
Understanding sample size is crucial when it comes to statistical confidence intervals. In essence, sample size refers to the number of observations or data points collected in a study. When estimating properties of a larger population based on a sample, having a sufficiently large sample size helps ensure that the sample adequately represents the population.

In statistical terms, the rule of thumb for the sample size when using the confidence interval for proportions requires that both the products, \( n\hat{p} \) and \( n(1 - \hat{p}) \), be at least 10. This ensures the normal approximation is valid, allowing for more accurate interval estimations.

When the sample size is too small, the sample might not accurately reflect the population's true characteristics, which can lead to inaccurate or misleading confidence intervals.
Proportion Estimation
Proportion estimation is a statistical technique used to approximate the ratio of a particular property within a population. For instance, if you want to estimate the proportion of individuals in a town who enjoy biking, you would collect sample data and estimate the proportion based on this sample.

The formula \( \hat{p} \pm (z \text{ critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \) is used to calculate the confidence interval around the estimated sample proportion \( \hat{p} \). The critical value is usually found using a standard normal distribution table, corresponding to the desired confidence level, such as 95%.

This interval helps in understanding how much certainty we can have about the population proportion being close to our sample proportion.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental statistical principle that states when you have a sufficiently large sample size, the sampling distribution of the sample mean (or proportion) will approach a normal distribution, regardless of the distribution of the original data.

It's the backbone for justifying the use of normal-based confidence intervals, like those for proportion estimation. This theorem allows us to assume a normal distribution for the sample mean or proportion, making it easier to calculate probabilities and confidence intervals.

In practical terms, this means that if the sample size is large enough, the properties of normal distribution like symmetry around the mean and the standard deviation characteristics can be leveraged to make estimates about the population with a known degree of confidence.

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Most popular questions from this chapter

Based on data from a survey of 1200 randomly selected Facebook users (USA TODAY, March 24, 2010), a \(98 \%\) confidence interval for the proportion of all Facebook users who say it is OK to ignore a coworker's "friend" request is \((0.35,0.41) .\) What is the meaning of the confidence level of \(98 \%\) that is associated with this interval?

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8 ?\)

In a survey on supernatural experiences, 722 of 4013 adult Americans reported that they had seen a ghost ("What Supernatural Experiences We've Had," USA TODAY, February 8,2010 ). Assume that this sample is representative of the population of adult Americans. a. Use the given information to estimate the proportion of adult Americans who would say they have seen a ghost. b. Verify that the conditions for use of the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in context. e. Construct and interpret a \(90 \%\) confidence interval for the proportion of all adult Americans who would say they have seen a ghost. f. Would a \(99 \%\) confidence interval be narrower or wider than the interval calculated in Part (e)? Justify your answer.

Suppose that a city planning commission wants to know the proportion of city residents who support installing streetlights in the downtown area. Two different people independently selected random samples of city residents and used their sample data to construct the following confidence intervals for the population proportion: Interval 1:(0.28,0.34) Interval 2:(0.31,0.33) (Hint: Consider the formula for the confidence interval given on page 444.) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals have a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

In mid-2016 the United Kingdom (UK) withdrew from the European Union (an event known as "Brexit"), causing economic concerns throughout the world. One indicator that economists use to monitor the health of the economy is the proportion of residential properties offered for sale at auction that are successfully sold. An article titled "Going, going, gone through the roof-sky's the limit at auction" (October \(22,2016,\) www.estateagenttoday .co.uk/features/2016/10/going-going-gone-through-theroof-the-skys-the-limit- at-auction, retrieved May 4,2017 ) reported the success rate of a sample of 26 residential properties offered for sale at auctions in the UK in the summer of \(2016 .\) For this sample of properties, 14 of the 26 residential properties were successfully sold. Suppose it is reasonable to consider these 26 properties as representative of residential properties offered at auction in the post-Brexit UK. a. Would it be appropriate to use the large-sample confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. b. Would it be appropriate to use a bootstrap confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. c. Use the accompanying output from the "Bootstrap Confidence Interval for One Proportion" Shiny app to report a \(95 \%\) bootstrap confidence interval for the population proportion of residential properties successfully
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