Federal law might eventually specify that new automobiles must average, for exam
ID: 3200559 • Letter: F
Question
Federal law might eventually specify that new automobiles must average, for example, 45 miles per gallon (mpg) of gasoline. Because it’s impossible to test all new cars, compliance tests would be based on random samples from the entire production of each car model. If a hypothesis test indicates substandard performance, the manufacturer would be penalized, we’ll assume, $200 per car for the entire production. In these tests, the null hypothesis states that with respect to the mandated mean of 45 mpg, nothing special is happening in the population for some car model—that is, there is no substandard performance and the population mean equals or exceeds 45 mpg. The alternative hypothesis refl ects a concern that the population mean is less than 45 mpg. Symbolically, the two statistical hypotheses read:
H0: m $ 45
H1: m , 45
From the manufacturer’s perspective, a type I error (a stiff penalty, even though the car complies with the standard) is very serious. Accordingly, to control the type I error, let’s use the .01 instead of the customary .05 level of signifi cance. From the federal regulator’s perspective, a type II error (not penalizing the manufacturer even though the car fails to comply with the standard) also is serious. In practice, a sample size should be selected, as described in Section 11.11, to control the type II error—that is, to ensure a reasonable detection rate for the smallest decline (judged to be important) of the true population mean below the mandated 45 mpg. To simplify computations in the present example, however, the projected one-tailed test is based on data from a very small sample of only six randomly selected cars.
13.9 In the gas mileage test described in this chapter, would you prefer a smaller or a larger sample size if you were
(a) the car manufacturer? Why?
(b) a vigorous prosecutor for the federal regulatory agency? Why?
Explanation / Answer
Solution :-
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: >= 45
Alternative hypothesis: < 45
Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a one-sample t-test.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = s / sqrt(n) = 1 / sqrt(6) = 1/2.449 = 0.408
DF = 6 - 1 = 6 - 1 = 5
t = (x - ) / SE = (44 - 45 )/2.449 = -0.408
where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.
Here is the logic of the analysis: Given the alternative hypothesis ( < 45), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.
The observed sample mean produced a t statistic test statistic of 0.408. We use the t Distribution Calculator to find P(t < 0.408) = 0.350079.
The P-Value is 0.350079.
The result is not significant at p < 0.01.
Interpret results. Since the P-value (0.35) is greater than the significance level (0.01), we cannot reject the null hypothesis.
In the gas mileage test described in this chapter, would you prefer a smaller or a larger sample size if you were
(a) the car manufacturer? Why?
Sample size is important because
Larger samples increase your chance of significance is because they more reliably reflect the population mean.
Ideally the larger samples reflects the best information about the population but as it incurrs a hugh cost with it, mostly tests rely on ramdom selection of few samples.
Being a car manufacturer, I would prefer a vice selection of few samples (not to large) in order to cut down the cost, however keeping the other conditions in control.
(b) a vigorous prosecutor for the federal regulatory agency? Why?
As a vigorous prosecutor for the fedral regulatory agency, I will prefer a large sample from the population so as to attain the best reflect of the population. Being a procecutor, cost reduction is not a concern but getting the most accurate result is.
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