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Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. a. Define the random variables \(X\) and \(P^{\prime}\) in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 97\(\%\) confidence interval for the population proportion of people over 50 who ran and died in the same eight-year period. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. d. Explain what a 97\% confidence interval means for this study.

Step by step solution

01

Define Random Variables

Let \( X \) be the random variable representing the number of people over 50 who ran and died over the eight-year period. \( P^{\prime} \) represents the sample proportion of people over 50 who ran and died during the study period.

02

Choose Distribution

We should use the binomial distribution to model \( X \) since it represents the number of successes (deaths) in a fixed number of trials (451 individuals), where each trial has two possible outcomes (death or no death). More specifically, since we are dealing with proportions and a large sample size, we can use the normal approximation to the binomial distribution to construct the confidence interval.

03

Calculate the Sample Proportion

The sample proportion \( \hat{p} \) is calculated as follows: \( \hat{p} = \frac{1.5}{100} = 0.015 \), representing that 1.5% of the sample died within the study period.

04

Calculate the Confidence Interval

To construct a 97% confidence interval, we use the formula for confidence intervals with a normal approximation: \[ \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( Z \) is the z-value for 97%, \( \hat{p} = 0.015 \), and \( n = 451 \).

05

Determine Z-value for 97% Confidence Level

The Z-value for a 97% confidence interval can be found using a standard normal distribution table or calculator, which corresponds to approximately 2.17.

06

Calculate the Error Bound

Calculate the error bound using the formula: \[ E = Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 2.17 \times \sqrt{\frac{0.015(1-0.015)}{451}} \approx 0.010 \]

07

State the Confidence Interval

The 97% confidence interval is calculated as: \[ 0.015 \pm 0.010 \] which gives a lower bound of 0.005 and an upper bound of 0.025. So, the interval is \( (0.005, 0.025) \).

08

Explain the Meaning of Confidence Interval

A 97% confidence interval for this study means that we are 97% confident the true population proportion of people over 50 who ran and died in the same eight-year period falls between 0.5% and 2.5%.

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Key Concepts

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

Confidence Interval

A confidence interval is a statistical tool used to estimate a range within which a certain population parameter lies. In our case, it is used to estimate the true proportion of people over 50 who ran and died within an eight-year study period.

Confidence intervals are expressed as "X% confident that the true parameter falls between two values". For example, a 97% confidence interval suggests that if the study were repeated multiple times, 97% of the time the interval would contain the true population parameter.

Confidence intervals provide a way to gauge the uncertainty or precision of our estimate. They are particularly useful because they consider sample variation and can give us insight into the reliability of our results.

Normal Approximation

Normal approximation is an approach used for simplifying certain calculations in statistical analysis, particularly when working with distributions like the binomial distribution.

In scenarios with a large sample size, the binomial distribution can be closely approximated by a normal distribution, thanks to the Central Limit Theorem. This is beneficial because calculations involving normal distributions tend to be easier due to their symmetrical, bell-shaped curve.

In our study, where we are concerned with deaths among 451 individuals, the use of a normal approximation simplifies the process of creating a confidence interval for the sample proportion, as we essentially rely on previously established z-values, such as the 2.17 used here.

Binomial Distribution

The binomial distribution is essential for modeling situations where there are a fixed number of independent trials, each resulting in a success or a failure. In our study, these are the instances of death among runners over 50.

The binomial distribution is defined by two parameters: the number of trials (n, which is 451 in our study) and the probability of success (p, here 0.015). Such distribution gives a discrete probability of a certain number of successes in these trials.

Understanding whether a scenario fits the criteria for a binomial distribution is key as it determines the most appropriate statistical methods for analysis, particularly in estimating proportions and constructing confidence intervals.

Sample Proportion

The sample proportion provides an estimate of the population proportion and is calculated by taking the ratio of observed successes to total trials. For our exercise, it was determined by dividing the number of people who died by the total sample size, calculated as 0.015 or 1.5%.

Sample proportions allow researchers to effortlessly summarize studies, especially large ones, and can be used to draw inferences about the larger population from which the sample is drawn.

In our context, grasping the concept of a sample proportion is crucial for constructing the confidence interval, because it acts as a foundation for understanding the broader study population and the expected variation between different samples.