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Chapter 9: Q11 E (page 372)

The level of lead in the blood was determined for a sample of \(152\) male hazardous-waste workers ages \(20 - 30\) and also for a sample of \(86\) female workers, resulting in a mean \(6\) standard error of \(5.5 \pm 0.3\)for the men and \(3.8 \pm 0.2\) for the women (鈥淭emporal Changes in Blood Lead Levels of Hazardous Waste Workers in New Jersey, 1984鈥�1987,鈥� Environ. Monitoring and Assessment, 1993: 99鈥�107). Calculate an estimate of the difference between true average blood lead levels for male and female workers in a way that provides information about reliability and precision.

Short Answer

Expert verified

the solution is \({\rm{(0}}{\rm{.99,2}}{\rm{.41)}}\)

Step by step solution

01

calculate the difference between average blood level male and female workers

The confidence interval for \({\mu _1} - {\mu _2}\) with a confidence level of around \(100(1 - \alpha )\% \) percent is for \(m\), \(n\)large enough.

\(\bar x - \bar y \pm {z_{\alpha /2}} \times \sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} \)

where \( + \) and \( - \) denote the interval's appropriate upper and lower limits. An upper/lower bound is obtained by substituting \({z_{\alpha /2}}\) with \({z_\alpha }\) and \( \pm \)with only \( + \) and \( - \).

Because both sample sizes are large enough, the confidence interval specified before can be employed. The stated information regarding the first sample is in the exercise.

\(\begin{array}{l}\bar x = 5.5\\S{E_1} = \frac{{s_1^2}}{m} = 0.{3^2}\end{array}\)

And about the second sample are

\(\begin{array}{l}\bar y = 3.8\\S{E_2} = \frac{{s_2^2}}{n} = 0.{2^2}\end{array}\)

Using confidence level of \(\alpha = 0.05\), the \( z\) value is \({z_{\alpha /2}} = {z_{0.025}} = 1.96\). The \( 95\% \)confidence interval becomes

\(\begin{array}{l}\bar x - \bar y \pm {z_{\alpha /2}} \times \sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} = 5.5 - 3.8 \pm 1.96 \times \sqrt {0.{3^2} + 0.{2^2}} \\ = (0.99,2.41).\end{array}\)

The genuine average blood lead level for male workers is somewhere between \({\rm{0}}{\rm{.99}}\) and greater than the average for female workers, according to \(95\) confidence.

02

conclusion

the final solution is

\({\rm{(0}}{\rm{.99,2}}{\rm{.41)}}\)

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Most popular questions from this chapter

The article "Flexure of Concrete Beams Reinforced with Advanced Composite Orthogrids"\((J\). of Aerospace Engr., 1997: 7-15) gave the accompanying data on ultimate load\((kN)\)for two different types of beams.

\( - 7.0944\)

a. Assuming that the underlying distributions are normal, calculate and interpret a\(99\% \)CI for the difference between true average load for the fiberglass beams and that for the carbon beams.

b. Does the upper limit of the interval you calculated in part (a) give a\(99\% \)upper confidence bound for the difference between the two\(\mu \)'s? If not, calculate such a bound. Does it strongly suggest that true average load for the carbon beams is more than that for the fiberglass beams? Explain.

Researchers sent 5000 resumes in response to job ads that appeared in the Boston Globe and Chicago Tribune. The resumes were identical except that 2500 of them had "white sounding" first names, such as Brett and Emily, whereas the other 2500 had "black sounding" names such as Tamika and Rasheed. The resumes of the first type elicited 250 responses and the resumes of the second type only 167 responses (these numbers are very consistent with information that appeared in a Jan. 15. 2003, report by the Associated Press). Does this data strongly suggest that a resume with a "black" name is less likely to result in a response than is a resume with a "white" name?

In an experiment to compare bearing strengths of pegs inserted in two different types of mounts, a sample of 14 observations on stress limit for red oak mounts resulted in a sample mean and sample standard deviation of \(8.48MPa\) and .79 MPa, respectively, whereas a sample of 12 observations when Douglas fir mounts were used gave a mean of \(9.36\) and a standard deviation of \(1.52\) ('Bearing Strength of White Oak Pegs in Red Oak and Douglas Fir Timbers," J. of Testing and Evaluation, 1998, 109-114). Consider testing whether or not true average stress limits are identical for the two types of mounts. Compare df's and P-values for the unpooled and pooled t tests.

An experiment was performed to compare the fracture toughness of high-purity \(18Ni\) maraging steel with commercial-purity steel of the same type (Corrosion Science, 1971: 723鈥�736). For \(m = 32\)specimens, the sample average toughness was \(\overline x = 65.6\) for the high purity steel, whereas for \(n = 38\)specimens of commercial steel \(\overline y = 59.8\). Because the high-purity steel is more expensive, its use for a certain application can be justified only if its fracture toughness exceeds that of commercial purity steel by more than 5. Suppose that both toughness distributions are normal.

a. Assuming that \({\sigma _1} = 1.2\) and \({\sigma _2} = 1.1\), test the relevant hypotheses using \(\alpha = .001\).

b. Compute \(\beta \) for the test conducted in part (a) when \({\mu _1} - {\mu _2} = 6.\)

Sometimes experiments involving success or failure responses are run in a paired or before/after manner. Suppose that before a major policy speech by a political candidate, n individuals are selected and asked whether \((S)\)or not (F) they favor the candidate. Then after the speech the same n people are asked the same question. The responses can be entered in a table as follows:

Before

After

S

F

S

\({{\bf{X}}_{\bf{1}}}\)

\({{\bf{X}}_{\bf{2}}}\)

F

\({{\bf{X}}_{\bf{3}}}\)

\({{\bf{X}}_{\bf{4}}}\)

Where\({{\bf{x}}_{\bf{1}}}{\bf{ + }}{{\bf{x}}_{\bf{2}}}{\bf{ + }}{{\bf{x}}_{\bf{3}}}{\bf{ + }}{{\bf{x}}_{\bf{4}}}{\bf{ = n}}\). Let\({{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}\), and \({p_4}\)denote the four cell probabilities, so that \({p_1} = P(S\) before and S after), and so on. We wish to test the hypothesis that the true proportion of supporters (S) after the speech has not increased against the alternative that it has increased.

a. State the two hypotheses of interest in terms of\({p_1},{p_2}\),\({p_3}\), and \({p_4}\).

b. Construct an estimator for the after/before difference in success probabilities

c. When n is large, it can be shown that the rv \(\left( {{{\bf{X}}_{\bf{i}}}{\bf{ - }}{{\bf{X}}_{\bf{j}}}} \right){\bf{/n}}\) has approximately a normal distribution with variance given by\(\left[ {{{\bf{p}}_{\bf{i}}}{\bf{ + }}{{\bf{p}}_{\bf{j}}}{\bf{ - }}{{\left( {{{\bf{p}}_{\bf{i}}}{\bf{ - }}{{\bf{p}}_{\bf{j}}}} \right)}^{\bf{2}}}} \right]{\bf{/n}}\). Use this to construct a test statistic with approximately a standard normal distribution when \({H_0}\)is true (the result is called McNemar's test).

d. If\({{\bf{x}}_{\bf{1}}}{\bf{ = 350,}}\;\;\;{{\bf{x}}_{\bf{2}}}{\bf{ = 150,}}\;\;\;{{\bf{x}}_{\bf{3}}}{\bf{ = 200}}\), and\({x_4} = 300\), what do you conclude?
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