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Chapter 7: Problem 3

In order to estimate the proportion \(p\) of people who own houses in a district, we choose a random sample from the population and study its sampling distribution. Assuming \(p=0.3,\) use the appropriate formulas from this section to find the mean and the standard deviation of the sampling distribution of the sample proportion for a random sample of size: a. \(n=400\). b. \(n=1600\). c. \(n=100\). d. Summarize the effect of the sample size on the size of the standard deviation.

Short Answer

Expert verified
The mean is 0.3 for all, with standard deviations: 0.0229 (n=400), 0.0115 (n=1600), 0.0458 (n=100). Larger samples decrease standard deviation.

Step by step solution

01

Understanding the Mean of the Sampling Distribution

The mean of the sampling distribution of the sample proportion \( p \) is equal to the population proportion \( p \). Given that \( p = 0.3 \), the mean \( \mu_{\hat{p}} \) is also \( 0.3 \).
02

Standard Deviation Formula for Sampling Distribution

The standard deviation of the sampling distribution of the sample proportion \( \hat{p} \) is given by the formula: \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \). We will use this formula to calculate the standard deviation for different sample sizes.
03

Calculate Standard Deviation for n=400

Substitute \( p = 0.3 \) and \( n = 400 \) into the standard deviation formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.3 \times 0.7}{400}} = \sqrt{\frac{0.21}{400}} = \sqrt{0.000525} \approx 0.0229 \].
04

Calculate Standard Deviation for n=1600

Substitute \( p = 0.3 \) and \( n = 1600 \) into the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.3 \times 0.7}{1600}} = \sqrt{\frac{0.21}{1600}} = \sqrt{0.00013125} \approx 0.0115 \].
05

Calculate Standard Deviation for n=100

Substitute \( p = 0.3 \) and \( n = 100 \) into the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.3 \times 0.7}{100}} = \sqrt{\frac{0.21}{100}} = \sqrt{0.0021} \approx 0.0458 \].
06

Effect of Sample Size on Standard Deviation

As the sample size \( n \) increases, the standard deviation of the sample proportion \( \sigma_{\hat{p}} \) decreases. This is because the formula \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \) includes \( n \) in the denominator, thus larger \( n \) results in a smaller \( \sigma_{\hat{p}} \).

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Key Concepts

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

Sample Proportion
When we want to estimate how many people in a district own houses, we often focus on a sample proportion. The sample proportion is denoted as \( \hat{p} \) and represents the fraction of the sampled individuals who exhibit the characteristic we're interested in, such as owning a house.
To calculate the sample proportion, you divide the number of individuals with the characteristic by the total number of individuals in the sample.
  • If 120 out of 400 people in your sample own a house, your sample proportion \( \hat{p} \) would be \( \frac{120}{400} = 0.3 \).
This reflected fraction helps in understanding and predicting the behavior of the whole population, especially when dealing with large groups.
Our primary goal is to make inferences about the entire population based on observing the sample proportion.
Standard Deviation
The standard deviation of a sampling distribution is a measure of how spread out the values are around the mean. In simpler terms, it tells us how much the sample proportions can vary.In the context of sample proportions, the standard deviation formula is \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \). This equation is key for understanding the distribution's variability:
  • \( p \) is the population proportion, e.g., the fraction of the entire district population that owns houses.
  • \( n \) is the sample size, i.e., the number of individuals surveyed.
To put it simply, a smaller standard deviation means your sample proportions closely cluster around the true value of the population proportion. This makes predictions more reliable. A larger standard deviation indicates more variability in your sample proportions.
Population Proportion
Population proportion, denoted by \( p \), represents a true fraction of the entire population that possesses a certain characteristic, like owning a house. In our example, if 30% of the district owns houses, then the population proportion, \( p \), is 0.3.
  • The population proportion is often unknown, so researchers rely on sample surveys to estimate it.
  • This estimated figure helps make data-driven decisions, predicts trends, and assesses areas that may require attention or resources.
It is important to note that the population proportion remains constant across different samples, while the sample proportion may differ because of random sampling variability.
Sample Size Effect
Understanding the effect of sample size is crucial in statistics. The sample size, \( n \), directly impacts the standard deviation of the sample proportion. A larger sample size can provide more accurate estimates of the population proportion.
  • When \( n \) is small, the standard deviation \( \sigma_{\hat{p}} \) is larger, implying greater variability in sample proportion values. This can result in less reliable estimates.
  • As \( n \) grows, \( \sigma_{\hat{p}} \) becomes smaller. This tightens the range in which you would expect to find the sample proportions. In plain words, there's less deviation from the population proportion.
Therefore, increasing the sample size enhances the precision of the proportion estimation, allowing more accurate and reliable predictions about the population's behavior.

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Most popular questions from this chapter

The sampling distribution of a sample mean for a random sample size of 100 describes a. How sample means tend to vary from random sample to random sample of size 100 . b. How observations tend to vary from person to person in a random sample of size 100 . c. How the data distribution looks like the population distribution when the sample size is larger than 30 . d. How the standard deviation varies among samples of size 100 .

Based on data from the 2010 Major League Baseball season, \(X=\) number of home runs the San Francisco Giants hit in a game has a mean of 1.0 and a standard deviation of 1.0 . a. Do you think \(X\) has a normal distribution? Why or why not? b. Suppose that this year \(X\) has the same distribution. Report the shape, mean, and standard deviation of the sampling distribution of the mean number of home runs the team will hit in its 162 games. c. Based on the answer to part b, find the probability that the mean number of home runs per game in this coming season will exceed 1.50 .

Each student should bring 10 coins to class. For each coin, observe its age, the difference between the current year and the year on the coin. a. Using all the students' observations, the class should construct a histogram of the sample ages. What is its shape? b. Now each student should find the mean for that student's 10 coins, and the class should plot the means of all the students. What type of distribution is this, and how does it compare to the one in part a? What concepts does this exercise illustrate?

The central limit theorem implies a. All variables have approximately bell-shaped data distributions if a random sample contains at least about 30 observations. b. Population distributions are normal whenever the population size is large. c. For sufficiently large random samples, the sampling distribution of \(\bar{x}\) is approximately normal, regardless of the shape of the population distribution. d. The sampling distribution of the sample mean looks more like the population distribution as the sample size increases.

In a class of 150 students, the professor has each person toss a fair coin 50 times and calculate the proportion of times the tosses come up heads. Roughly \(95 \%\) of students should have proportions between which two numbers? a. 0.49 and 0.51 b. 0.05 and 0.95 c. 0.42 and 0.58 d. 0.36 and 0.64 e. 0.25 and 0.75 Explain your answer.
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