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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 10: Q. 95 (page 690)

There are two common methods for measuring the concentration of a pollutant in fish tissue. Do the two methods differ, on average? You apply both methods to each fish in a random sample of $18$ carp and use
a. the paired t test for $μ d i f f \frac{305}{1526} = 0.200 = 20.0 % μ d i f f$ .
b. the one-sample z test for p.
c. the two-sample t test for $μ 1 - μ 2 \frac{305}{1526} = 0.200 = 20.0 % μ 1 - μ 2$ .
d. the two-sample z test for $p 1 - p 2 \frac{305}{1526} = 0.200 = 20.0 % p 1 - p 2$ .
e. none of these.

Short Answer

Expert verified

The correct option is:

a. the paired t test for $μ d i f f \frac{305}{1526} = 0.200 = 20.0 % μ d i f f$

Step by step solution

01

Step 1 - Given information:

We have been given that:

Fish in a random sample of $18$ crap.

02

Step 2 - Explanation:

Generally:

One percentage z-test/interval: one sample

Two proportions z-test/interval: two samples

a case study one mean, t-test/interval

There are two possibilities: on two samples t-test/interval

Using two samples of data, two means or mean difference: t-test/interval paired t-test

If you're looking for a difference, quality, rise, or decline, a test is a good way to go.

Use an interval to approximate the real value's range.

Because both procedures are used on the same sample, the paired t-test is used.

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Most popular questions from this chapter

Literacy A researcher reports that $80 %$ of high school graduates, but only $40 %$ of high school dropouts, would pass a basic literacy test. Assume that the researcher’s claim is true. Suppose we give a basic literacy test to a random sample of $60$ high school graduates and a separate random sample of $75$ high school dropouts. ${\hat{p}}_{G}, {\hat{p}}_{D}$ be the sample proportions of graduates and dropouts, respectively, who pass the test.

a. What is the shape of the sampling distribution of ${\hat{p}}_{G} - {\hat{p}}_{D}$ . Why?

b. Find the mean of the sampling distribution.

c. Calculate and interpret the standard deviation of the sampling distribution.

Better barley Does drying barley seeds in a kiln increase the yield of barley? A famous experiment by William S. Gosset (who discovered the t distributions) investigated this question. Eleven pairs of adjacent plots were marked out in a large field. For each pair, regular barley seeds were planted in one plot and kiln-dried seeds were planted in the other. A coin flip was used to determine which plot in each pair got the regular barley seed and which got the kiln-dried seed. The following table displays the data on barley yield (pound per acre) for each plot.

Do these data provide convincing evidence at the $α = 0.05$ level that drying barley seeds in a kiln increases the yield of barley, on average?

Researchers suspect that Variety A tomato plants have a different average yield than Variety B tomato plants. To find out, researchers randomly select $10$ Variety A and $10$ Variety B tomato plants. Then the researchers divide in half each of $10$ small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety B plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The $10$ differences (Variety A − Variety B) in yield are recorded. A graph of the differences looks roughly symmetric and single-peaked with no outliers. The mean difference is $\bar{x} = 0.34$ $\frac{305}{1526} = 0.200 = 20 % {\bar{x}}_{A - B} = 0.34$ and the standard deviation of the differences is s A-B $= 0.83$ $\frac{305}{1526} = 0.200 = 20 % = s_{A - B} = 0.83$ .Let $μ_{A - B} = \frac{305}{1526} = 0.200 = 20 %$ μA−B = the true mean difference (Variety A − Variety B) in yield for tomato plants of these two varieties.

The P-value for a test of H0: μA−B=0 $\frac{305}{1526} = 0.200 = 20 %$ versus Ha: μA−B≠0 is 0.227. Which of the following is the

correct interpretation of this P-value?

a. The probability that μA−B is $0.227$ .

b. Given that the true mean difference (Variety A – Variety B) in yield for these two varieties of tomato plants is $0$ , the probability of getting a sample mean difference of $0.34$ is $0.227$ .

c. Given that the true mean difference (Variety A – Variety B) in yield for these two varieties of tomato plants is $0$ , the probability of getting a sample mean difference of $0.34$ or greater is $0.227$ .

d. Given that the true mean difference (Variety A – Variety B) in yield for these two varieties of tomato plants is $0$ , the probability of getting a sample mean difference greater than or equal to $0.34$ or less than or equal to − $0.34$ is $0.227$ .

e. Given that the true mean difference (Variety A – Variety B) in yield for these two varieties of tomato plants is not 0, the probability of getting a sample mean difference greater than or equal to $0.34$ or less than or equal to − $0.34$ is $0.227$ .

American-made cars Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but they live in different states. In Nathan’s state, $80 %$ of the registered cars are made by American manufacturers. In Kyle’s state, only $60 %$ of the registered cars are made by American manufacturers. Nathan selects a random sample of $100$ cars in his state and Kyle selects a random sample of $70$ cars in his state. Let $p^{n} - p^{k}$ be the difference (Nathan’s state – Kyle’s state) in the sample proportion of cars made by American manufacturers.

a. What is the shape of the sampling distribution of $p^{n} - p^{k}$ ? Why?

b. Find the mean of the sampling distribution.

c. Calculate and interpret the standard deviation of the sampling distribution.

On your mark In track, sprinters typically use starting blocks because they think it will help them run a faster race. To test this belief, an experiment was designed where each sprinter on a track team ran a $50$ -meter dash two times, once using starting blocks and once with a standing start. The order of the two different types of starts was determined at random for each sprinter. The times (in seconds) for 8 different sprinters are shown in the table.

a. Make a dotplot of the difference (Standing - Blocks) in $50$ -meter run time for each sprinter. What does the graph suggest about whether starting blocks are helpful?

b. Calculate the mean difference and the standard deviation of the differences. Explain why the mean difference gives some evidence that starting blocks are helpful.

c. Do the data provide convincing evidence that sprinters like these run a faster race when using starting blocks, on average?

d. Construct and interpret a $90 %$ confidence interval for the true mean difference. Explain how the confidence interval gives more information than the test in part (b).

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