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Chapter 16: Problem 2

Marketing The proportion of adult women in the United States is approximately \(51 \%\). A marketing survey telephones 400 people at random. a. What proportion of the sample of 400 would you expect to be women? b. What would the standard deviation of the sampling distribution be? c. How many women, on average, would you expect to find in a sample of that size?

Short Answer

Expert verified
The expected proportion of women in the sample is 0.51. The standard deviation of the sampling distribution would be calculated using the formula: \(\sqrt{0.51 * (1 - 0.51) / 400}\). On average, one would expect to find around 204 women in a sample of this size.

Step by step solution

01

Calculate Expected Proportion

First use the given proportion of adult women in the U.S, which is \(51\%\). Convert this to a decimal by dividing by 100. Hence, the expected proportion of women in a sample of 400 will be: \(0.51 = 51\% / 100\). Thus, in a random sample of 400 people, \(51\%\) are expected to be women.
02

Calculate Standard Deviation

Next, calculate the standard deviation of the sampling distribution. In statistics, the standard deviation for a proportion is calculated using the formula: \(\sqrt{p(1 - p) / n}\), where p is the population proportion, and n is the sample size. Thus, for this problem, the standard deviation will be: \(\sqrt{0.51(1 - 0.51) / 400}\)
03

Calculate Expected Number of Women

Finally, use the proportion to find the expected number of women in a sample of 400. To do this, multiply the sample size by the proportion: \(400 * 0.51\). This calculation indicates that, on average, about 204 women are expected to be found in a sample of this size.

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Key Concepts

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

Proportion
A proportion is simply a way of expressing a part of a whole. It's often expressed as a decimal or a percentage, representing what fraction of a group exhibits a particular characteristic.
In our context, we are talking about the proportion of adult women in a given population. With the exercise in focus, the population consists of people in the United States, and 51% of them are adult women. This means, in any group or sample you take from this population, you would expect around 51% to be women.
  • It's a practical tool in statistics for making predictions about a sample based on known population data.
  • Helps to understand the demographic makeup of a group.
  • Aids in estimating outcomes based on given data.
In real-world scenarios, proportions help in planning, decision-making, and predicting future trends based on current data.
Standard Deviation
Standard deviation is a measure of how much variation or spread exists from the average (mean) in a set of data.
When looking at proportions in statistics, the standard deviation tells us how much we can expect individual sample proportions to deviate from the population proportion. For proportion problems, the formula is:\[ \sqrt{p(1-p)/n} \]Where:
  • \(p\) is the population proportion.
  • \(1-p\) is the proportion not having the characteristic.
  • \(n\) is the sample size.
Using this formula, you can estimate the variation in your sample's results. If you have a small standard deviation, your sample proportions are close to the population proportion, indicating lower variability. This concept is crucial in validating survey results, ensuring sample data accurately reflects the overall population.
Population Proportion
In sampling distribution, the population proportion is the ratio of members in a group that have a specific attribute or characteristic relative to the whole population.
In our example, the population proportion of adult women is 0.51 or 51%. This means every time we randomly pick people from the U.S. population, about 51% should be women, on average.
  • Essential for calculating the expected number of occurrences in a sample.
  • Acts as a benchmark for determining how well a sample represents the larger group.
  • Helps to construct confidence intervals and perform hypothesis testing.
The population proportion is foundational in enabling statisticians and researchers to draw insights from data while understanding the general population's characteristics.
Sample Size
Sample size refers to the number of observations or data points collected from a population to conduct statistical analysis.
A well-chosen sample size is crucial because it determines the accuracy of predictions and conclusions drawn from an analysis. In our exercise, the sample size is 400.
  • Larger sample sizes typically lead to more reliable and stable statistical estimates.
  • Increase the likelihood of representing the population accurately.
  • Can reduce the margin of error in predictions or extrapolations.
Selecting an appropriate sample size is vital, especially in surveys and experiments, as it helps balance resources and data accuracy effectively.

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Most popular questions from this chapter

Another pilot study During routine screening, a doctor notices that \(22 \%\) of her adult patients show higher than normal levels of glucose in their blood-a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within \(4 \%\) with \(98 \%\) confidence. How many randomly selected adults must they test?

Conclusions A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a \(95 \%\) confidence interval for the proportion of all orders that arrive on time is \(88 \% \pm 6 \%\). What does this mean? Are these conclusions correct? Explain. a. Between \(82 \%\) and \(94 \%\) of all orders arrive on time. b. Ninety-five percent of all random samples of customers will show that \(88 \%\) of orders arrive on time. c. Ninety-five percent of all random samples of customers will show that \(82 \%\) to \(94 \%\) of orders arrive on time. d. We are \(95 \%\) sure that between \(82 \%\) and \(94 \%\) of the orders placed by the sampled customers arrived on time. e. On \(95 \%\) of the days, between \(82 \%\) and \(94 \%\) of the orders will arrive on time.

Send money When they send out their fundraising letters, a philanthropic organization typically gets a return from about \(5 \%\) of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size \(20,50,100,\) and 200 . For each sample size, they simulated 1000 mailings with success rate \(p=0.05\) and constructed the histogram of the 1000 sample proportions, shown below. Explain what these histograms show about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

Campus sample For her final project, Stacy plans on surveying a random sample of 50 students on whether they plan to go to Florida for spring break. From past years, she guesses that about \(10 \%\) of the class goes. Is it reasonablefor her to use a Normal model for the sampling distribution of the sample proportion? Why or why not?

31\. Mislabeled seafood In 2013 the environmental group Oceana (usa.oceana.org) analyzed 1215 samples of seafood purchased across the United States and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(33 \%\) of the seafood was mislabeled according to U.S. Food and Drug Administration guidelines. a. Construct a \(95 \%\) confidence interval for the proportion of all seafood sold in the United States that is mislabeled or misidentified. b. Explain what your confidence interval says about seafood sold in the United States. c. A 2009 report by the Government Accountability Office says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 1215 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.
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