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Chapter 8: Problem 11

Five years ago \(3.9 \%\) of children in a certain region lived with someone other than a parent. A sociologist wishes to test whether the current proportion is different. Perform the relevant test at the \(5 \%\) level of significance using the following data: in a random sample of 2,759 children, 119 lived with someone other than a parent.

Short Answer

Expert verified
With a calculated z-value exceeding 1.96, the proportion has changed from 3.9%.

Step by step solution

01

Identify the Hypotheses

First, we need to establish our null and alternative hypotheses. The null hypothesis (\(H_0\)) is that the proportion of children currently living with someone other than a parent is still \(3.9\%\) or \(p = 0.039\). The alternative hypothesis (\(H_a\)) is that the proportion is different from \(3.9\%\), so \(p eq 0.039\).
02

Collect and Analyze the Sample Data

In a sample of 2,759 children, 119 are living with someone other than a parent. Calculate the sample proportion \( \hat{p} \) as follows: \( \hat{p} = \frac{119}{2759} \approx 0.0431 \).
03

Calculate the Test Statistic

To test the hypothesis, use the formula for the test statistic for a proportion: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \), where \( p_0 = 0.039 \) and \( n = 2759 \). Substitute the values: \( z = \frac{0.0431 - 0.039}{\sqrt{\frac{0.039 \times 0.961}{2759}}} \). Calculate to get the test statistic value.
04

Determine the Critical Value or p-value

At a \(5\%\) level of significance for a two-tailed test, the critical z-values are approximately \( \pm 1.96 \). You can also calculate the p-value using the test statistic in a Z-table or with statistical software to further support your findings.
05

Make a Decision

Compare the test statistic to the critical values. If the absolute value of the test statistic is greater than \(1.96\), we reject the null hypothesis. Otherwise, we do not reject the null hypothesis. Similarly, if the p-value is less than \(0.05\), reject the null hypothesis.
06

Conclusion

Based on the comparison if necessary, conclude whether there is enough statistical evidence to say the proportion has changed from \(3.9\%\).

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Key Concepts

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

Proportion Hypothesis
In hypothesis testing, setting up the hypotheses is a critical step. For proportion hypothesis tests, this involves determining a null and an alternative hypothesis. The null hypothesis ( H_0 ) represents the status quo or a baseline assumption. When we say the proportion is 3.9%, it essentially implies there's no change.
  • The null hypothesis ( H_0 ) for our case was: the current proportion of children living with someone other than a parent is still 3.9%, or p = 0.039 .
  • The alternative hypothesis ( H_a ), on the other hand, questions this status quo and suggests a change. For a two-tailed test, it is that the proportion is not 3.9%, written as p 鈮� 0.039 .
Recognizing these hypotheses is foundational, as your entire test revolves around proving or refuting the null hypothesis based on statistical evidence.
Significance Level
The significance level, often denoted as \(\alpha\), is a crucial part of hypothesis testing. It sets the probability threshold for rejecting a true null hypothesis. In simpler terms, it's your maximum permissible chance of making a mistake.
  • A significance level of \(0.05\), or 5%, is commonly used, as in our example.
  • This means there's a 5% risk of concluding a difference exists when there isn't one, effectively a 5% chance of making a Type I error.
      • This threshold helps in determining your critical values, where you'll see whether the null hypothesis deserves rejection. Setting a proper significance level is crucial because it strikes a balance between Type I and Type II errors.
Z-Test
The Z-test is a statistical method used to determine if there are differences between sample and population proportions. It's applicable when your data meets specific criteria, like a large sample size.
  • To perform a Z-test for a proportion, you calculate a Z-statistic using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \]
  • Here, \( \hat{p} \) is the sample proportion, \( p_0 \) is the population proportion, and \( n \) is the sample size.
In practice, once you compute the Z-value, you compare it with critical values from the standard normal distribution. A value beyond these critical points suggests the sample proportion is significantly different, leading you to reject the null hypothesis. The Z-test is practical for hypothesis testing due to its simplicity and ease of understanding.
Sample Proportion
The sample proportion is a key figure in hypothesis testing when you're dealing with proportions. It provides an estimate of the population proportion based on your sample.
  • The sample proportion, denoted \(\hat{p}\), is calculated by dividing the number of successes (children living with non-parents here) by the total sample size.
  • For instance, in our exercise, it was \(\hat{p} = \frac{119}{2759} \approx 0.0431\).
This figure is crucial not just for your Z-test calculation, but also for understanding how close or far your sample is from the assumed population proportion. A big difference between \(\hat{p}\) and \(p_0\) might suggest a significant change has occurred within the population.

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Most popular questions from this chapter

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two- tailed. a. \(\quad H 0: \mu=141\) VS. Ha: \(\mu<141\) \(@ \alpha=0.20\). b. \(\quad H 0: \mu=-54\) vs. Ha: \(\mu<-54 @ \alpha=0.05\). C. \(\quad H 0: \mu=98.6\) VS. \(H a: \mu \neq 98.6 @ \alpha=0.05 .\) d. \(\quad H 0: \mu=3.8\) VS. Ha: \(\mu>3.8 @ \alpha=0.001\).

The mean yield for hard red winter wheat in a certain state is 44.8 bu/acre. In a pilot program a modified growing scheme was introduced on 35 independent plots. The result was a sample mean yield of 45.4 bu/acre with sample standard deviation 1.6 bu/acre, an apparent increase in yield. a. Test at the \(5 \%\) level of significance whether the mean yield under the new scheme is greater than 44.8 bu/acre, using the critical value approach. b. Compute the observed significance of the test. c. Perform the test at the \(5 \%\) level of significance using the \(p\) -value approach. You need not repeat the first three steps, already done in part (a).

Perform the indicated test of hypotheses, based on the information given. a. Test \(\mathrm{Ho}: \mu=212\) vs. Ha: \(\mu<212 @ \alpha=0.10, \sigma\) unknown, \(n=36, x-211.2, \mathrm{~s}=2.2\) b. Test \(H_{0}: \mu=-18\) vs. Ha: \(\mu>-18\) \(@ \alpha=0.05, \sigma=3.3, n=44, x-=-17.2, s=3.1\) c. Test \(H_{0}: \mu=24\) vs. Ha: \(\mu \neq 24 @ \alpha=0.02, \sigma\) unknown, \(n=50, x-=22.8, s=1.9\)

Large Data Set 6 records results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate \(A\) for a U.S. Senate seat or prefer some other candidate. Use the full data set (400 observations) to test the hypothesis that the proportion \(p\) of all voters who prefer Candidate \(A\) exceeds \(0.35 .\) Test at the \(10 \%\) level of significance.

According to the Federal Poverty Measure \(12 \%\) of the U.S. population lives in poverty. The governor of a certain state believes that the proportion there is lower. In a sample of size 1,550,163 were impoverished according to the federal measure. a. Test whether the true proportion of the state's population that is impoverished is less than \(12 \%,\) at the \(5 \%\) level of significance. 1\. Compute the observed significance of the test.
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