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Chapter 9: Problem 61

The \(z\) statistic for a test of \(H_{0}: p=0.4\) versus \(H_{a}: p \neq 0.4\) is \(z=2.43 .\) This test is (a) not significant at either \(\alpha=0.05\) or \(\alpha=0.01\). (b) significant at \(\alpha=0.05\) but not at \(\alpha=0.01\). (c) significant at \(\alpha=0.01\) but not at \(\alpha=0.05\). (d) significant at both \(\alpha=0.05\) and \(\alpha=0.01\). (e) inconclusive because we don't know the value of \(\hat{p}\).

Short Answer

Expert verified
(b) significant at \( \alpha=0.05 \) but not at \( \alpha=0.01 \).

Step by step solution

01

Identify the critical values

For a two-tailed test with significance level \( \alpha = 0.05 \), we look up the critical values from the standard normal \( z \)-table. These critical values are approximately \( z = \pm 1.96 \). For \( \alpha = 0.01 \), the critical values are approximately \( z = \pm 2.576 \).
02

Compare the test statistic to critical values

The given test statistic is \( z = 2.43 \). Compare this with the critical values at both significance levels:- For \( \alpha = 0.05 \), \(2.43 > 1.96\), so the test statistic is in the critical region.- For \( \alpha = 0.01 \), \(2.43 < 2.576\), so the test statistic is not in the critical region.
03

Decide significance

Since the test statistic \( z = 2.43 \) is within the critical region for \( \alpha = 0.05 \) but not for \( \alpha = 0.01 \), the test is significant at \( \alpha = 0.05 \) but not at \( \alpha = 0.01 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Z-Statistic
The z-statistic is a key concept in hypothesis testing. It represents the number of standard deviations a data point is from the mean. In simple terms, the z-statistic measures how extreme a test statistic is when compared to the null hypothesis assumption. For instance, in our problem, the z-statistic is 2.43. This value tells us how far away our sample proportion is from the hypothesized proportion under the null hypothesis, assuming the data follows a standard normal distribution. The larger the absolute value of the z-statistic, the more evidence there is against the null hypothesis. Understanding the z-statistic involves:
  • Calculating how a sample deviates from the population mean under the null hypothesis.
  • Interpreting this deviation in terms of standard normal distribution.
Importance of the Significance Level
The significance level, denoted by \( \alpha \), is a threshold used to decide whether a test result is statistically significant. Common levels are 0.05 and 0.01, which correspond to 5% and 1% respectively. At these levels, we define how confident we are in rejecting the null hypothesis.Choosing a significance level involves balancing the risk of Type I error (rejecting a true null hypothesis). A lower \( \alpha \) means you're more stringent, thus requiring stronger evidence to reject the null. In our exercise, different \( \alpha \) values determine whether our z-statistic falls into the critical region or not, influencing the decision of the test mechanism.
Critical Values in Hypothesis Testing
Critical values are the boundaries that separate the critical region from the non-critical region in a distribution. They are crucial in hypothesis testing because they help us decide whether to reject the null hypothesis.For a two-tailed test at \( \alpha = 0.05 \), the critical values are \( \pm 1.96 \), while for \( \alpha = 0.01 \), they are \( \pm 2.576 \). These values are derived from the standard normal distribution.Comparing the test statistic to these critical values determines its significance. If the test statistic falls beyond the critical values, it lies in the critical region, suggesting significant results against the null hypothesis.
Explaining a Two-Tailed Test
A two-tailed test is used when we are interested in deviations in either direction from the hypothesized parameter. In this test, the alternative hypothesis indicates that the population proportion is not equal (either higher or lower) to a specified value.This means that we are considering both extremes in the distribution: too high or too low relative to the mean. Therefore, the critical region encompasses both tails of the distribution. In our case, we have used a two-tailed test with critical values at both \( +z \) and \( -z \), reflecting the possibility of deviations in either direction away from the hypothesized mean. This approach ensures that we catch any significant differences, regardless of their direction.

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Most popular questions from this chapter

A manufacturer of compact discs (CDs) wants to be sure that their CDs will fit inside the plastic cases they have bought for packaging. Both the CDs and the cases are circular. According to the supplier, the plastic cases vary Normally with mean diameter \(\mu=4.2\) inches and standard deviation \(\sigma=0.05\) inches. The CD manufacturer decides to produce CDs with mean diameter \(\mu=4\) inches. Their diameters follow a Normal distribution with \(\sigma=0.1\) inches. (a) Let \(X=\) the diameter of a randomly selected \(\mathrm{CD}\) and \(Y=\) the diameter of a randomly selected case. Describe the shape, center, and spread of the distribution of the random variable \(X-Y\). What is the importance of this random variable to the CD manufacturer? (b) Compute the probability that a randomly selected CD will fit inside a randomly selected case. (c) The production process actually runs in batches of 100 CDs. If each of these CDs is paired with a randomly chosen plastic case, find the probability that all the CDs fit in their cases.

In Exercises 7 to 10, explain what's wrong with the stated hypotheses. Then give correct hypotheses. A change is made that should improve student satisfaction with the parking situation at a local high school. Right now, \(37 \%\) of students approve of the parking that's provided. The null hypothesis \(H_{0}: p>0.37\) is tested against the alternative \(H_{a}: p=0.37\)

In the sample, \(\hat{p}=158 / 300=0.527 .\) The resulting \(P\) -value is 0.18 . What is the correct interpretation of this \(P\) -value? (a) Only \(18 \%\) of the city residents support the tax increase. (b) There is an \(18 \%\) chance that the majority of residents supports the tax increase. (c) Assuming that \(50 \%\) of residents support the tax increase, there is an \(18 \%\) probability that the sample proportion would be 0.527 or higher by chance alone. (d) Assuming that more than \(50 \%\) of residents support the tax increase, there is an \(18 \%\) probability that the sample proportion would be 0.527 or higher by chance alone. (e) Assuming that \(50 \%\) of residents support the tax increase, there is an \(18 \%\) chance that the null hypothesis is true by chance alone.

After once again losing a football game to the archrival, a college's alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic is (a) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}}\) (b) \(t=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}}\) (c) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}\) (d) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{64}}}\) (e) \(z=\frac{0.5-0.64}{\sqrt{\frac{0.5(0.5)}{100}}}\)

Are boys more likely? We hear that newborn babies are more likely to be boys than girls. Is this true? \(\mathrm{A}\) random sample of 25,468 firstborn children included 13,173 boys. (a) Do these data give convincing evidence that firstborn children are more likely to be boys than girls? (b) To what population can the results of this study be generalized: all children or all firstborn children? Justify your answer.
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