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Chapter 6: Problem 1

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion, find the standard error of the distribution of sample proportions. Samples of size 50 from a population with proportion 0.25

Short Answer

Expert verified
The standard error for a sample proportion of a sample of size 50 from a population with a proportion of 0.25 is approximately \(0.069\).

Step by step solution

01

Identify the required parameters

The given population proportion \(p\) is 0.25, the sample size \(n\) is 50. Calculate the complement \(q\) which is 1 - \(p\), hence \(q\) equals 0.75.
02

Use the standard error formula

Substitute the identified values into the formula \(SE = \sqrt{pq/n}\). Therefore, \(SE = \sqrt{(0.25 * 0.75) / 50}\).
03

Calculate the standard error

Perform the multiplication and the division, and then square root the result to obtain the standard error.

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Key Concepts

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

Sample Proportion
The sample proportion is a key concept when dealing with statistics and probability. It represents the fraction of individuals in a sample that have a particular characteristic of interest. In simple terms, it is an estimate of the proportion in the entire population based on what is observed in a smaller sample.

When you look at a sample, you might notice that not everyone in the sample shares a certain trait. By dividing the number of individuals with that trait by the total number of individuals in the sample, you get the sample proportion. For instance, if you have 50 ballplayers and 20 are pitchers, the sample proportion of pitchers is 20 out of 50, or 0.4.
  • Sample Proportion formula: \(\hat{p} = \frac{x}{n}\)
  • Where \(x\) is the count of successes in the sample, and \(n\) is the sample size.
This allows statisticians to make educated guesses about the population without needing to study everyone.
Population Proportion
The population proportion is the true proportion of a certain trait across the entire population. Unlike the sample proportion, which is an estimate, the population proportion is considered to be an exact figure鈥攂ut it鈥檚 often unknown.

We use symbols like \(p\) to denote this value. In our problem, the population proportion \(p\) is provided as 0.25. This means that in the whole population, 25% possess the trait of interest.
  • Population Proportion (\(p\)) helps determine expectations for the whole group.
  • Used as a benchmark to compare the sample proportion with what could be expected from the entire population.
Understanding this difference is critical, as it helps in analyzing how well the sample mirrors real-world conditions.
Sample Size
Sample size, denoted as \(n\), plays a crucial role in statistics because it directly affects the accuracy of your results. Larger sample sizes tend to yield more reliable estimates of the population parameters because they reduce random errors.

In our example, the sample size \(n\) is 50. So, 50 individuals or items are randomly selected from the population to represent it. The larger the sample size, the closer the sample proportion is expected to be to the population proportion.
  • A larger \(n\) generally means more accurate results.
  • Sample size calculation forms the basis for computing the standard error, which helps quantify the uncertainty in the sample's representation of the population.
It's vital to choose an appropriate sample size to ensure that the findings are statistically significant.

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Most popular questions from this chapter

Football Air Pressure During the NFL's 2014 AFC championship game, officials measured the air pressure on game balls following a tip that one team's balls were under-inflated. In exercise 6.124 we found that the 11 balls measured for the New England Patriots had a mean psi of 11.10 (well below the legal limit) and a standard deviation of 0.40. Patriot supporters could argue that the under-inflated balls were due to the elements and other outside effects. To test this the officials also measured 4 balls from the opposing team (Indianapolis Colts) to be used in comparison and found a mean psi of \(12.63,\) with a standard deviation of 0.12. There is no significant skewness or outliers in the data. Use the t-distribution to determine if the average air pressure in the New England Patriot's balls was significantly less than the average air pressure in the Indianapolis Colt's balls.

Use a t-distribution and the given matched pair sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distribution of the differences is relatively normal. Assume that differences are computed using \(d=x_{1}-x_{2}\). Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2}\) using the paired difference sample results \(\bar{x}_{d}=15.7, s_{d}=12.2\) \(n_{d}=25 .\)

In Exercises 6.152 and \(6.153,\) find a \(95 \%\) confidence interval for the difference in proportions two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the normal distribution and the formula for standard error. Compare the results. Difference in proportion who use text messaging, using \(\hat{p}_{t}=0.87\) with \(n=800\) for teens and \(\hat{p}_{a}=0.72\) with \(n=2252\) for adults.

Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the proportion in a t-distribution above 2.1 if the samples have sizes \(n_{1}=12\) and \(n_{2}=12\).

A data collection method is described to investigate a difference in means. In each case, determine which data analysis method is more appropriate: paired data difference in means or difference in means with two separate groups. In a study to determine whether the color red increases how attractive men find women, one group of men rate the attractiveness of a woman after seeing her picture on a red background and another group of men rate the same woman after seeing her picture on a white background.
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