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Chapter 8: Problem 4

True or False: The mean of the sampling distribution of \(\hat{p}\) is \(p\).

Short Answer

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True

Step by step solution

01

Understand the Problem

Determine if the statement 'The mean of the sampling distribution of \( \hat{p} \) is \( p \)' is true or false. To do this, review the properties of the sampling distribution of a sample proportion.
02

Review Concept of Sampling Distribution

Recall that the sampling distribution of the sample proportion \( \hat{p} \) describes how the proportion \( \hat{p} \) calculated from different samples varies. One of its key properties is that the mean of the sampling distribution of \( \hat{p} \) equals the true population proportion \( p \).
03

Check the Mean of \( \hat{p} \)

Using the property mentioned, confirm that the mean of the sampling distribution of \( \hat{p} \) is indeed \( p \). This is a fundamental result in statistics which holds true.
04

Conclusion

Since the mean of the sampling distribution of \( \hat{p} \) is \( p \), the statement given is true.

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Key Concepts

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

Sample Proportion
In statistics, a sample proportion, denoted by \( \hat{p} \), is simply the fraction or percentage of individuals in a sample with a specific characteristic. For instance, if you're surveying a group of people about their voting preferences and 60 out of 100 favor a particular candidate, the sample proportion is \( \hat{p} = \frac{60}{100} = 0.60 \). When you take different samples from the same population, the sample proportions can vary. This variability forms the basis of what we call a 'sampling distribution'. Understanding sample proportion is crucial because it helps in making inferences about the population from which the sample is drawn.
Mean of Sampling Distribution
The concept of the 'mean of the sampling distribution' is fundamental in statistics. When we take multiple samples from a population and calculate a specific statistic (e.g., sample proportion) for each sample, the collection of these statistics forms a 'sampling distribution'. The mean of this sampling distribution is simply the average of these statistics. According to a key property of sampling distributions, the mean of the sampling distribution of the sample proportion \(\hat{p} \) is equal to the population proportion \( p \). This means that if you repeatedly take samples and calculate the sample proportion, the average of these sample proportions will converge to the true population proportion. This principle validates why sample estimates are generally unbiased estimators of population parameters.
Population Proportion
The 'population proportion', denoted by \( p \), refers to the fraction or percentage of the entire population that possesses a certain characteristic. For example, if 20% of a city鈥檚 population are joggers, then the population proportion of joggers is \( p = 0.20 \). This proportion is often unknown and estimated through sample surveys. One of the important aspects of population proportion is its influence on the properties of sampling distributions. When we draw random samples and calculate the sample proportion \(\hat{p} \) multiple times, the average of these calculated proportions gives us an unbiased estimate of \( p \). Understanding population proportion is key to making reliable statistical inferences from samples to broader populations.

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Most popular questions from this chapter

A simple random sample of size \(n=40\) is obtained from a population with \(\mu=50\) and \(\sigma=4 .\) Does the population need to be normally distributed for the sampling distribution of \(\bar{x}\) to be approximately normally distributed? Why? What is the sampling distribution of \(\bar{x} ?\)

Finite Population Correction Factor In this section, we assumed that the sample size was less than \(5 \%\) of the size of the population. When sampling without replacement from a finite population in which \(n>0.05 N,\) the standard deviation of the distribution of \(\hat{p}\) is given by $$ \sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n-1} \cdot\left(\frac{N-n}{N}\right)} $$ where \(N\) is the size of the population. A survey is conducted at a college having an enrollment of 6502 students. The student council wants to estimate the percentage of students in favor of establishing a student union. In a random sample of 500 students, it was determined that 410 were in favor of establishing a student union. (a) Obtain the sample proportion, \(\hat{p},\) of students surveyed who favor establishing a student union. (b) Calculate the standard deviation of the sampling distribution of \(\hat{p}\) using \(\hat{p}\) as an estimate of \(p\).

A simple random sample of size \(n=36\) is obtained from a population with \(\mu=64\) and \(\sigma=18\). (a) Describe the sampling distribution of \(\bar{x}\). (b) What is \(P(\bar{x}<62.6) ?\) (c) What is \(P(\bar{x} \geq 68.7) ?\) (d) What is \(P(59.8<\bar{x}<65.9) ?\)

Suppose you want to study the number of hours of sleep you get each evening. To do so, you look at the calendar and randomly select 10 days out of the next 300 days and record the number of hours you sleep. (a) Explain why number of hours of sleep in a night by you is a random variable. (b) Is the random variable "number of hours of sleep in a night" quantitative or qualitative? (c) After you obtain your ten nights of data, you compute the mean number of hours of sleep. Is this a statistic or a parameter? Why? (d) Is the mean number of hours computed in part (c) a random variable? Why? If it is a random variable, what is the source of variation?

Fumbles The New England Patriots made headlines prior to the 2015 Super Bowl for allegedly playing with underinflated footballs. An underinflated ball is easier to grip, and therefore, less likely to be fumbled. What does the data say? The following data represent the number of plays a team has per fumble. For example, the Chicago Bears run 48 offensive plays for every fumble. $$ \begin{array}{llc|llc} \text { Team } & \text { Dome } & \text { Plays } & \text { Team } & \text { Dome } & \text { Plays } \\ \hline \text { Atlanta Falcons } & \text { Yes } & 83 & \text { Saint Louis Rams } & \text { Yes } & 50 \\ \hline \text { New Orleans Saints } & \text { Yes } & 80 & \text { Seattle Seahawks } & \text { No } & 50 \\ \hline \text { New England Patriots } & \text { No } & 78 & \text { Detroit Lions } & \text { Yes } & 49 \\ \hline \text { Houston Texans } & \text { Yes } & 61 & \text { Chicago Bears } & \text { No } & 48 \\ \hline \text { Minnesota Vikings } & \text { Yes } & 59 & \text { San Francisco 49ers } & \text { No } & 47 \\ \hline \text { Baltimore Ravens } & \text { No } & 58 & \text { New York Jets } & \text { No } & 46 \\ \hline \text { Carolina Panthers } & \text { No } & 57 & \text { Denver Broncos } & \text { No } & 46 \\ \hline \text { San Diego Chargers } & \text { No } & 57 & \text { Tampa Bay Buccaneers } & \text { No } & 45 \\ \hline \text { Indianapolis Colts } & \text { Yes } & 56 & \text { Dallas Cowboys } & \text { Yes } & 45 \\ \hline \text { Green Bay Packers } & \text { No } & 55 & \text { Tennessee Titans } & \text { No } & 45 \\ \hline \text { New York Giants } & \text { No } & 55 & \text { Oakland Raiders } & \text { No } & 44 \\ \hline \text { Cincinnati Bengals } & \text { No } & 53 & \text { Miami Dolphins } & \text { No } & 44 \\ \hline \text { Pittsburgh Steelers } & \text { No } & 52 & \text { Buffalo Bills } & \text { No } & 43 \\ \hline \text { Kansas City Chiefs } & \text { No } & 52 & \text { Arizona Cardinals } & \text { Yes } & 43 \\ \hline \text { Jacksonville Jaguars } & \text { No } & 51 & \text { Philadelphia Eagles } & \text { No } & 42 \\ \hline \text { Cleveland Browns } & \text { No } & 50 & \text { Washington Redskins } & \text { No } & 37 \\ \hline \end{array} $$ (a) Draw a boxplot of plays per fumble for all teams in the National Football League. Describe the shape of the distribution. Are there any outliers? If so, which team(s)? (b) Playing in a dome (inside) removes the effect of weather (such as rain) on the game. Draw a boxplot of plays per fumble for teams who do not play in a dome. Are there any outliers? If so, which team(s)?
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