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Chapter 8: Problem 77

A study is done to see whether a coin is biased. The alternative hypothesis used is two-sided, and the obtained \(z\) -value is 2 . Assuming that the sample size is sufficiently large and that the other conditions are also satisfied, use the Empirical Rule to approximate the p-value.

Short Answer

Expert verified
The p-value is approximately 0.025.

Step by step solution

01

Identifying critical z-values

First, since this is a two-sided test, it means that the critical region is both below and above the hypothesized mean. For a two-tailed test with a significance level of \( \alpha = 0.05 \), the critical z-values are \(-1.96\) and \(1.96\). If the observed z-value falls in the critical region, then the null hypothesis is rejected.
02

Comparing z-values

The obtained \(z\)-value is 2 which is greater than 1.96. This means that the observed z-value falls in the critical region, so the null hypothesis of the coin being unbiased is rejected.
03

Approximating the p-value using Empirical Rule

Given that the z-value is 2, and knowing that about 95% of the data falls within two standard deviations (z = 卤1.96) of the mean according to the Empirical Rule, the portion of the data that's more than 2 standard deviations from the mean is about 5% total for both tails. Since we are in a two-tailed situation, we split this 5% into half to get 2.5% in each tail (.025 when expressed as a proportion). This gives an approximate p-value of 0.025.

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Most popular questions from this chapter

Choosing a Test and Naming the Population(s) In each case, choose whether the appropriate test is a one-proportion z-test or a two-proportion z-test. Name the population(s). a. A person is observed to check whether he or she can predict the results of the die roll better than chance alone. b. A rescarcher wants to know whether a new skin cream effectively reduces skin rashes compared to an old cream. c. A survey agency conducts a survey in a city to find out whether the residents like dark and white chocolate equally. d. A news agency takes a random sample of all corporate professionals to see whether more than \(50 \%\) approve of the new tax regime. e. A teacher takes a random sample of students in statistics class to find out whether girls or boys are more likely to remember the conditions for applying \(z\) -test.

In 2012 the Ventura Coumty Star reported that \(77 \%\) of employers allow employees to use flex time and periodically change their start and quit times (this is up from \(66 \%\) in 2005 ). Suppose a random sample of 200 employers shows that 130 allow flex time. Test the hypothesis that the percentage is less than \(77 \%\), using a significance level of \(0.10\).

For each of the following, state whether a one-proportion z-test or a two- proportion z-test would be appropriate, and name the populations. The minimum quantity of ingredient \(\mathrm{X}\) in a product is \(38 \%\) of total weight. A random sample of products manufactured in a factory is examined to see whether the rate of ingredient \(\mathrm{X}\) in the product is significantly higher than \(38 \%\). b. A student watches a random sample of male and female car drivers parking in a parking lot. Some drivers park the car as they drive in and some park after reversing the car. He wants to compare the proportions of male and female drivers who park the car after reversing.

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. Suppose a poll is taken, and 580 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. If a confidence interval is appropriate, find the approximate \(95 \%\) confidence interval. In both cases, assume that the necessary conditions have been met.

Mercury in Freshwater Fish (Example 11) Some experts believe that \(20 \%\) of all freshwater fish in the United States have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 250 fish tested, and 60 of them have dangerous levels of mercury. Test the hypothesis that this sample is not from a population with \(20 \%\) dangerous fish. Use a significance level of \(0.05\). Comment on your conclusion: Are you saying that the percentage of dangerous fish is definitely \(20 \%\) ? Explain.
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