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Chapter 6: Problem 5

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 300 from a population with proportion 0.08

Short Answer

Expert verified
The standard error of the distribution of sample proportions from a population with proportion 0.08 and sample size of 300 is obtained by substituting these values into the standard error formula and performing the calculation.

Step by step solution

01

Identify the values

Firstly, the values of the proportion \((p)\) and the sample size \((n)\) from the problem need to be identified. From the problem, it can be seen that the values are \(p = 0.08\) and \(n = 300\).
02

Substitute the values into the formula

Next, substitute the values into the standard error formula. Thus it becomes, \(\sqrt{\frac{0.08(1 - 0.08)}{300}}\).
03

Perform the calculation

Finally, perform the given calculation to find the standard error. Use any scientific calculator for this process.

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Most popular questions from this chapter

In Exercises 6.34 to 6.36, we examine the effect of different inputs on determining the sample size needed to obtain a specific margin of error when finding a confidence interval for a proportion. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within \(\pm 6 \%\) when estimating a proportion. Within \(\pm 4 \%\). Within \(\pm 1 \%\). (Assume no prior knowledge about the population proportion \(p\).) Comment on the relationship between the sample size and the desired margin of error.

Is Gender Bias Influenced by Faculty Gender? Exercise 6.215 describes a study in which science faculty members are asked to recommend a salary for a lab manager applicant. All the faculty members received the same application, with half randomly given a male name and half randomly given a female name. In Exercise \(6.215,\) we see that the applications with female names received a significantly lower recommended salary. Does gender of the evaluator make a difference? In particular, considering only the 64 applications with female names, is the mean recommended salary different depending on the gender of the evaluating faculty member? The 32 male faculty gave a mean starting salary of \(\$ 27,111\) with a standard deviation of \(\$ 6948\) while the 32 female faculty gave a mean starting salary of \(\$ 25,000\) with a standard deviation of \(\$ 7966 .\) Show all details of the test.

In Exercises 6.32 and 6.33, find a \(95 \%\) confidence interval for the proportion 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. Proportion of home team wins in soccer, using \(\hat{p}=0.583\) with \(n=120\)

In Exercises 6.150 and \(6.151,\) use StatKey or other technology to generate a bootstrap distribution of sample differences in proportions and find the standard error for that distribution. Compare the result to the value obtained using the formula for the standard error of a difference in proportions from this section. Sample A has a count of 30 successes with \(n=100\) and Sample \(\mathrm{B}\) has a count of 50 successes with \(n=250\).

\(\mathbf{6 . 2 2 0}\) Diet Cola and Calcium Exercise B.3 on page 349 introduces a study examining the effect of diet cola consumption on calcium levels in women. A sample of 16 healthy women aged 18 to 40 were randomly assigned to drink 24 ounces of either diet cola or water. Their urine was collected for three hours after ingestion of the beverage and calcium excretion (in \(\mathrm{mg}\) ) was measured. The summary statistics for diet cola are \(\bar{x}_{C}=56.0\) with \(s_{C}=4.93\) and \(n_{C}=8\) and the summary statistics for water are \(\bar{x}_{W}=49.1\) with \(s_{W}=3.64\) and \(n_{W}=8 .\) Figure 6.20 shows dotplots of the data values. Test whether there is evidence that diet cola leaches calcium out of the system, which would increase the amount of calcium in the urine for diet cola drinkers. In Exercise \(\mathrm{B} .3\), we used a randomization distribution to conduct this test. Use a t-distribution here, after first checking that the conditions are met and explaining your reasoning. The data are stored in ColaCalcium.
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