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Chapter 7: Problem 19

Calculate SE \((\hat{p})\) for \(n=100\) and the values of p given in Exercises \(16-22 .\) $$p=.50$$

Short Answer

Expert verified
Answer: The Standard Error of the proportion for a sample size of 100 and a proportion value of 0.50 is 0.05.

Step by step solution

01

Identify the given values

We are given the following values: - Sample size (\(n\)): 100 - Proportion (\(p\)): 0.50
02

Apply the formula for the Standard Error of a proportion

Use the formula for the Standard Error of a proportion which is: $$SE(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$$ Plug in the given values into the formula: $$SE(\hat{p}) = \sqrt{\frac{0.50(1-0.50)}{100}}$$
03

Simplify the expression

Simplify the expression inside the square root: $$SE(\hat{p}) = \sqrt{\frac{0.50(0.50)}{100}}$$
04

Calculate the Standard Error

Perform the calculation to find the Standard Error: $$SE(\hat{p}) = \sqrt{\frac{0.25}{100}}$$ $$SE(\hat{p}) = \sqrt{0.0025}$$ $$SE(\hat{p}) = 0.05$$ The Standard Error of the proportion \(\hat{p}\) for \(n=100\) and \(p=0.50\) is \(0.05\).

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Key Concepts

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

Probability and Statistics
Understanding the fundamentals of Probability and Statistics is like having a roadmap for navigating through data. One essential concept here is the Standard Error (SE) of a proportion, which plays a crucial role in describing how much we can expect sample proportions to vary from the population proportion. It鈥檚 the standard deviation of the sampling distribution of the proportion.

The calculation of SE helps statisticians to infer about the population, based on sample data. It鈥檚 all about uncertainty and variability. When you hear 'Standard Error,' think of it as a gauge of how spread out the proportions are in repeated random samples of a given size. The smaller the SE, the more certain we can be about the estimate of our population parameter 鈥� in this case, the population proportion.
Sample size (n)
The Sample size (n) is the number of observations in a sample and it's one of the main drivers in the precision of our estimates. In the context of our exercise, a sample size of 100 is relatively moderate, allowing some degree of reliability without requiring too much data collection.

When determining the SE of a proportion, the choice of n directly impacts our results. Larger samples tend to give more accurate estimates of the population parameter and result in a smaller SE. Why is that? Because as the sample size increases, the sample means cluster more closely around the population mean. Hence, the larger the n, the clearer the picture we get of the true population proportion, assuming our sample is representative.
Population proportion (p)
The Population proportion (p) represents the ratio of members in a population who have a particular attribute. It鈥檚 a key parameter in statistical analyses, especially when we're dealing with categorical binary data - think yes/no, success/failure, on/off, and so on.

In our exercise, p is set to 0.50, indicating a perfectly balanced scenario - as if to say half the population has the attribute in question, and the other half does not. Such a proportion often simplifies calculations but also holds specific importance in probability theory: it maximizes variability. If we were to plot the sampling distribution of the sample proportion with p = 0.50, it would take on a normal distribution shape more rapidly as the sample size increases, compared to other values of p.

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Most popular questions from this chapter

In a Fox News Poll \(^{3}\) conducted by Pulse Opinion Research in the state of Delaware, 1000 likely voters were asked to answer the following questions. \- All in all, would you rather have bigger government that provides more services or smaller government that provides fewer services? \- Do you agree or disagree with the following statement: The federal government has gotten totally out of control and threatens our basic liberties unless we clear house and commit to drastic change. \- Thinking about the health care law that was passed earlier this year, would you favor repealing the new law to keep it from going into effect, or would you oppose repealing the new law? Comment on the effect of wording bias on the responses gathered using this survey.

A gambling casino records and plots the mean daily gain or loss from five blackjack tables on an \(\bar{x}\) chart. The overall mean of the sample means and the standard deviation of the combined data over 40 weeks were \(\overline{\bar{x}}=\$ 10,752\) and \(s=\$ 1605,\) respectively. a. Construct an \(\bar{x}\) chart for the mean daily gain per blackjack table. b. How can this \(\bar{x}\) chart be of value to the manager of the casino?

A population consists of \(N=4\) numbers: 10,15,21,22 with a mean of \(\mu=17 .\) A random sample of \(n=2\) is selected in one of two ways: first, without replacement and second, with replacement. Use this information to answer the questions. How many possible random samples are available when sampling with replacement? List the possible samples. Find the sampling distribution for the sample mean \(\bar{x}\) and display it as a table and as a probability histogram.

The total daily sales, \(S\), in the deli section of a supermarket is the sum of the purchases made by customers on a given day. a. What kind of probability distribution do you expect the total daily sales to have? Explain. b. For this particular market, the average purchase per customer in the deli section is \(\$ 8.50\) with \(\sigma=\$ 2.50 .\) If 30 customers make deli purchases on a given day, give the mean and standard deviation of the probability distribution of the total daily sales, \(S\)

Does the race of an interviewer matter? This question was investigated and reported in an issue of Chance magazine. \(^{2}\) The interviewer asked, "Do you feel that affirmative action should be used as an occupation selection criteria?" with possible answers of yes or no. a. What problems might you expect with responses to this question when asked by interviewers of different ethnic origins? b. When people were interviewed by an African American, the response was about \(70 \%\) in favor of affirmative action, approximately \(35 \%\) when interviewed by an Asian, and approximately \(25 \%\) when interviewed by a Caucasian. Do these results support your answer in part a?
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