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Chapter 5: Problem 24

In each part below, there is a value of interest and two scenarios (I and II). For each part, report if the value of interest is larger under scenario I, scenario II, or whether the value is equal under the scenarios. (a) The standard error of \(\hat{p}\) when (I) \(n=125\) or (II) \(n=500\). (b) The margin of error of a confidence interval when the confidence level is (I) \(90 \%\) or (II) \(80 \%\). (c) The p-value for a Z-statistic of 2.5 calculated based on a (I) sample with \(n=500\) or based on a (II) sample with \(n=1000\). (d) The probability of making a Type 2 Error when the alternative hypothesis is true and the significance level is (I) 0.05 or (II) 0.10 .

Short Answer

Expert verified
(a) Larger in scenario I; (b) Larger in scenario I; (c) Equal; (d) Larger in scenario I.

Step by step solution

01

Standard Error of Proportion Formula

The standard error of the sample proportion \( \hat{p} \) is calculated using the formula: \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Here, the standard error is inversely proportional to the square root of \( n \).
02

Compare Standard Errors in Part (a)

For scenario (I), the sample size \( n = 125 \); for scenario (II), \( n = 500 \). Since the standard error is inversely proportional to the square root of \( n \), a larger \( n \) results in a smaller standard error. Thus, the standard error of \( \hat{p} \) is larger under scenario (I).
03

Margin of Error Formula

The margin of error for a confidence interval is calculated using the formula: margin of error = \( Z^{\ast} \times \text{SE} \). Where \( Z^{\ast} \) represents the critical value from the standard normal distribution that corresponds to the desired confidence level.
04

Compare Margins of Error in Part (b)

For scenario (I), the confidence level is \( 90\% \), resulting in a larger \( Z^{\ast} \) value than scenario (II), which has an \( 80\% \) confidence level. Thus, the margin of error is larger under scenario (I).
05

Understanding P-value Calculation

The p-value is determined by the Z-statistic and does not depend directly on the sample size. Thus, it remains constant for the same Z-statistic, irrespective of the sample size.
06

Compare P-values in Part (c)

Since the Z-statistic of 2.5 is the same for both scenarios (I) and (II), the p-value is equal under both scenarios.
07

Understanding Type 2 Error Probability

The probability of making a Type 2 error (failing to reject a false null hypothesis) is denoted by \( \beta \). This probability is affected by the significance level \( \alpha \), with a higher \( \alpha \) (less strict) typically resulting in a lower \( \beta \).
08

Compare Type 2 Error Probability in Part (d)

For scenario (I), \( \alpha = 0.05 \) and for scenario (II), \( \alpha = 0.10 \). A higher significance level (0.10) typically results in a lower \( \beta \). Therefore, the probability of making a Type 2 error is larger under scenario (I).

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Key Concepts

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

Standard Error
The standard error is a key concept in statistics that measures the variability or dispersion of a sample statistic, such as the sample mean or proportion, from the true population parameter. It is crucial in estimating how accurately the sample statistic represents the population.
  • Formula: The standard error (\( SE \)) of the sample proportion \( \hat{p} \) is determined using the formula: \[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. \]This formula shows that the standard error is inversely related to the sample size \( (n) \).
  • Impact of Sample Size: As the sample size \( n \) increases, the standard error decreases. This means that with a larger sample size, the variability of the estimate is reduced, leading to more reliable results.
Understanding standard error helps in assessing the precision of sample estimates and informs decisions about sample size requirements for studies.
Margin of Error
The margin of error provides a range around the sample statistic, indicating the precision of sample estimates and accounting for possible sampling errors. It is particularly relevant in the context of confidence intervals.
  • Formula: The margin of error is calculated as: \[ Margin \ of \ Error = Z^{\ast} \times SE \]where \( Z^{\ast} \) is the critical value reflecting the desired confidence level (e.g., 1.96 for 95% confidence).
  • Confidence Level: A higher confidence level will increase the critical value and thus, the margin of error. This indicates that estimates are less precise with higher confidence, reflecting increased uncertainty.
By understanding the margin of error, one can determine how much uncertainty is associated with sample estimates in statistical reports.
P-value
The p-value is a fundamental concept in hypothesis testing, used to determine the significance of results. It indicates the probability of observing the sample results, or more extreme ones, assuming that the null hypothesis is true.
  • Independence from Sample Size: Importantly, the p-value is computed from the Z-statistic and does not depend on the sample size. It relies entirely on the data and the statistical test being conducted.
  • Interpretation: A small p-value (typically less than 0.05) suggests that the observed data is unlikely under the null hypothesis, providing grounds to reject it. Conversely, a large p-value indicates a lack of evidence against the null hypothesis.
In practice, the p-value provides a measure to help decide on the validity of the initial hypothesis in the study.
Type 2 Error
Type 2 error, also known as beta (\( \beta \)), occurs when a statistical test fails to reject a false null hypothesis. This feature of hypothesis testing illustrates a lack of sensitivity in detecting real effects.
  • Relation to Significance Level: The likelihood of a Type 2 error is inversely related to the significance level (\( \alpha \)). A higher significance level, meaning less stringency, reduces the probability of a Type 2 error.
  • Balancing Errors: To achieve optimal testing conditions, statistical testing seeks a balance between Type 1 (false positive) and Type 2 errors, ensuring neither is excessively high.
Understanding Type 2 error is crucial for designing tests with adequate power to detect meaningful effects, while managing the risk of overlooking them.

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Most popular questions from this chapter

A nonprofit wants to understand the fraction of households that have elevated levels of lead in their drinking water. They expect at least \(5 \%\) of homes will have elevated levels of lead, but not more than about \(30 \%\). They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called? (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. (c) If the proportions are distributed around \(8 \%\), what is the variability of the distribution? (d) What is the formal name of the value you computed in (c)? (e) Suppose the researchers' budget is reduced, and they are only able to collect 250 observations per sample, but they can still collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 800 observations?

It is believed that nearsightedness affects about \(8 \%\) of all children. In a random sample of 194 children, 21 are nearsighted. Conduct a hypothesis test for the following question: do these data provide evidence that the \(8 \%\) value is inaccurate?

Suppose you conduct a hypothesis test based on a sample where the sample size is \(n=50,\) and arrive at a p-value of 0.08 . You then refer back to your notes and discover that you made a careless mistake, the sample size should have been \(n=500\). Will your p-value increase, decrease, or stay the same? Explain.

Exercise 5.11 provides a \(95 \%\) confidence interval for the mean waiting time at an emergency room (ER) of (128 minutes, 147 minutes). Answer the following questions based on this interval. (a) A local newspaper claims that the average waiting time at this ER exceeds 3 hours. Is this claim supported by the confidence interval? Explain your reasoning. (b) The Dean of Medicine at this hospital claims the average wait time is 2.2 hours. Is this claim supported by the confidence interval? Explain your reasoning. (c) Without actually calculating the interval, determine if the claim of the Dean from part (b) would be supported based on a \(99 \%\) confidence interval?

Determine whether the following statement is true or false, and explain your reasoning: "With large sample sizes, even small differences between the null value and the observed point estimate can be statistically significant."
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