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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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Most popular questions from this chapter

In Exercises 6.188 to 6.191 , use the t-distribution to find a confidence interval for a difference in means \(\mu_{1}-\mu_{2}\) given the relevant sample results. Give the best estimate for \(\mu_{1}-\mu_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples from populations that are approximately normally distributed. A \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the sample results \(\bar{x}_{1}=75.2, s_{1}=10.7, n_{1}=30\) and \(\bar{x}_{2}=69.0, s_{2}=8.3, n_{2}=20 .\)

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For each scenario, use the formula to find the standard error of the distribution of differences in sample means, \(\bar{x}_{1}-\bar{x}_{2}\) Samples of size 25 from Population 1 with mean 6.2 and standard deviation 3.7 and samples of size 40 from Population 2 with mean 8.1 and standard deviation 7.6

We saw in Exercise 6.221 on page 466 that drinking tea appears to offer a strong boost to the immune system. In a study extending the results of the study described in that exercise, \(^{58}\) blood samples were taken on five participants before and after one week of drinking about five cups of tea a day (the participants did not drink tea before the study started). The before and after blood samples were exposed to e. coli bacteria, and production of interferon gamma, a molecule that fights bacteria, viruses, and tumors, was measured. Mean production went from \(155 \mathrm{pg} / \mathrm{mL}\) before tea drinking to \(448 \mathrm{pg} / \mathrm{mL}\) after tea drinking. The mean difference for the five subjects is \(293 \mathrm{pg} / \mathrm{mL}\) with a standard deviation in the differences of \(242 .\) The paper implies that the use of the t-distribution is appropriate. (a) Why is it appropriate to use paired data in this analysis? (b) Find and interpret a \(90 \%\) confidence interval for the mean increase in production of interferon gamma after drinking tea for one week.

Use the t-distribution and the given 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 distributions are relatively normal. Test \(H_{0}: \mu_{A}=\mu_{B}\) vs \(H_{a}: \mu_{A} \neq \mu_{B}\) using the fact that Group A has 8 cases with a mean of 125 and a standard deviation of 18 while Group \(\mathrm{B}\) has 15 cases with a mean of 118 and a standard deviation of 14 .
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