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Chapter 6: Problem 1

Births There are about 11,000 births each day in the United States, and the proportion of boys born in the United States is \(0.512 .\) Assume that each day, 100 births are randomly selected and the proportion of boys is recorded. a. What do you know about the mean of the sample proportions? b. What do you know about the shape of the distribution of the sample proportions?

Short Answer

Expert verified
a. The mean is 0.512b. The shape is approximately normal.

Step by step solution

01

- Identify the Population Proportion

The population proportion of boys born in the United States is given as 0.512. This is denoted by the symbol \(\rho\).
02

- Determine the Mean of the Sample Proportions

According to the Central Limit Theorem, the mean of the sample proportions \(\bar{p}\) will be equal to the population proportion \(\rho\). Hence, the mean of the sample proportions is 0.512.
03

- Determine the Shape of the Distribution

The Central Limit Theorem also states that if the sample size is sufficiently large, the distribution of the sample proportions will be approximately normal. In this case, the sample size (n = 100) is large enough, so the distribution of the sample proportions will be approximately normal.

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Key Concepts

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

Population Proportion
In the context of the problem, the population proportion refers to the ratio of boys born in the United States each day. Given that the proportion of boys born is 0.512, this means that 51.2% of all births are boys. This value is denoted by the Greek letter \(\rho\). Understanding the population proportion is critical because it serves as a basis for comparing and analyzing sample proportions. It reflects the true, underlying probability in the entire population, giving us a standard to benchmark against.
Sample Proportions
When we take a sample from a population, such as recording the proportion of boys born among 100 randomly selected births, we're dealing with sample proportions. Unlike the population proportion, which remains fixed, sample proportions can vary from sample to sample. According to the Central Limit Theorem, the mean of the sample proportions \(\bar{p}\) tends to equal the population proportion \(\rho\). In our example, this means the average sample proportion of boys born should be around 0.512. This is important because it tells us that even though individual samples may vary, over many samples, the mean will align closely with the population proportion.
Normal Distribution
The shape of the distribution of sample proportions is particularly important. As per the Central Limit Theorem, if the sample size is sufficiently large (usually n > 30), the distribution of the sample proportions will approximate a normal distribution. In our exercise, the sample size is 100, which is large enough to satisfy this condition. This means the distribution of the sample proportions will resemble the bell-shaped curve of a normal distribution. Understanding this concept helps in making statistical inferences, as many statistical tests are based on the assumption of normality. This property allows us to make predictions and calculate probabilities effectively.

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Most popular questions from this chapter

The University of Maryland Medical Center considers "low birth weights" to be those that are less than 5.5 lb or 2495 g. Birth weights are normally distributed with a mean of \(3152.0 \mathrm{g}\) and a standard deviation of \(693.4 \mathrm{g}\) (based on Data Set 4 "Births" in Appendix B). a. If a birth weight is randomly selected, what is the probability that it is a "low birth weight"? b. Find the weights considered to be significantly low, using the criterion of a birth weight having a probability of 0.05 or less. c. Compare the results from parts (a) and (b).

After 1964, quarters were manufactured so that their weights have a mean of \(5.67 \mathrm{g}\) and a standard deviation of 0.06 g. Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters. a. If you adjust your vending machines to accept weights between \(5.60 \mathrm{g}\) and \(5.74 \mathrm{g}\). what percentage of legal quarters are rejected? Is that percentage too high? b. If you adjust vending machines to accept all legal quarters except those with weights in the top \(2.5 \%\) and the bottom \(2.5 \%,\) what are the limits of the weights that are accepted?

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Annual Incomes Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large \((n>30)\) random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is large? Why or why not?

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