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Chapter 9: Problem 58

\(\quad\) Consider \(H_{0}: p=.70\) versus \(H_{1}: p \neq .70\). a. A random sample of 600 observations produced a sample proportion equal to .68. Using \(\alpha=.01\), would you reject the null hypothesis? b. Another random sample of 600 observations taken from the same population produced a sample proportion equal to \(.76\). Using \(a=.01\), would you reject the null hypothesis? Comment on the results of parts a and b.

Short Answer

Expert verified
For the first case, we do not reject the null hypothesis. For the second case, we reject the null hypothesis.

Step by step solution

01

Compute the test statistic for the first sample.

First calculate the Z-score using the given sample proportion (p), hypothesized population proportion (p0), sample size(n), and standard error of proportion. The formula used is Z = (p - p0) / sqrt((p0*(1-p0))/n). Pew艂ace .68 for p, .70 for p0 and 600 for n. This gives us Z1 = -1.63.
02

Decision for the first sample

Using a significance level of 0.01, the critical Z-score for a two-tailed test are -2.58 and 2.58 (using standard normal distribution table). Compare Z1 with the critical Z-scores. Since -2.58 < -1.63 < 2.58, we do not reject the null hypothesis for the first case.
03

Compute the test statistic for the second sample.

We'll perform the same Z-score calculation for the second sample with p=.76. This gives us Z2 = 3.27.
04

Decision for the second sample

We compare Z2 with the critical Z-scores. Since 3.27 > 2.58, we reject the null hypothesis for the second case.
05

Comment on the results

For the first case (.68), the sample evidence isn't strong enough to reject the null hypothesis at the 1% level of significance. But for the second case (.76), the sample evidence suggests that the population proportion is significantly different from .70 at the 1% level of significance.

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Key Concepts

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

Z-Score Calculation
When testing hypotheses about population proportions, calculating the z-score is an important step. The z-score helps determine how far away the sample proportion is from the hypothesized population proportion, measured in units of standard error. The formula for calculating the z-score is:
  • \[ Z = \frac{p - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
Here, \(p\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and \(n\) is the sample size.
If the calculated z-score is far from zero, it suggests that there may be a significant difference between the sample and the hypothesized proportions.
Significance Level
The significance level, denoted by \(\alpha\), is a threshold for deciding if a statistical test result is significant. In this scenario, a 1% significance level (\(\alpha=0.01\)) is used. This implies that there is a 1% risk of concluding that a difference exists, when in fact, it doesn't.
Higher significance levels increase the chance of detecting a true effect, but also increase the risk of a type I error 鈥� falsely rejecting the null hypothesis. In summary, the significance level helps in deciding how strong the evidence must be before rejecting the null hypothesis.
Critical Values
Critical values divide the range of possible test statistics into regions where the null hypothesis is likely or unlikely. With a 1% significance level and a two-tailed test, the critical z-values are -2.58 and 2.58.
  • If the calculated z-score falls outside of these critical values, the null hypothesis is rejected.
  • If the z-score is within the critical values, we fail to reject the null hypothesis, suggesting the sample data isn't sufficiently unusual under the null hypothesis.
This decision-making process ensures that we are at least 99% confident in any conclusions made regarding population differences.
Sample Proportion
The sample proportion is a key concept in hypothesis testing, especially when dealing with categorical data. It is the fraction of the sample that displays a certain characteristic. Calculating the sample proportion involves dividing the number of observations with the characteristic of interest by the total sample size.
It provides an estimate of the true population proportion, and when combined with the sample size, gives us an idea of how reliable our estimate might be. The sample proportions in the original problem were 0.68 and 0.76, and their differences from the hypothesized value (\(p_0 = 0.70\)) were crucial in determining whether the null hypothesis should be rejected.

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Most popular questions from this chapter

Briefly explain the conditions that must hold true to use the \(t\) distribution to make a test of hypothesis about the population mean.

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.

More and more people are abandoning national brand products and buying store brand products to save money. The president of a company that produces national brand coffee claims that \(40 \%\) of the people prefer to buy national brand coffee. A random sample of 700 people who buy coffee showed that 259 of them buy national brand coffee. Using \(\alpha=.01\), can you conclude that the percentage of people who buy national brand coffee is different from \(40 \%\) ? Use both approaches to make the test.

According to Moebs Services Inc., the average cost of an individual checking account to major U.S. banks was \(\$ 380\) in 2013 (www. moebs.com). A bank consultant wants to determine whether the current mean cost of such checking accounts at major U.S. banks is more than \(\$ 380\) a year, A recent random sample of 150 such checking accounts taken from major U.S. banks produced a mean annual cost to them of \(\$ 390\). Assume that the standard deviation of annual costs to major banks of all such checking accounts is \(\$ 60 .\) a. Find the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, would you reject the null hypothesis if the maximum probability of Type I error is to be \(05 ?\) What if the maximum probability of Type I error is to be \(.01\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.05\). Would you reject the null hypothesis? What if \(\alpha=.01 ?\) What if \(a=0\) ?

According to the U.S. Census Bureau, in \(2014,62 \%\) of Americans age 18 and older were married. A recent sample of 2000 Americans age 18 and older showed that \(58 \%\) of them are married. Can you reject the null hypothesis at a \(1 \%\) significance level in favor of the alternative that the percentage of current population of Americans age 18 and older who are married is lower than \(62 \%\) ? Use both the \(p\) -value and the critical-value approaches.
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