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Chapter 11: Problem 12

Conduct each test at the a = 0.05 level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, (c) the critical value, and (d) the P-value. Assume that the samples were obtained independently using simple random sampling. Test whether \(p_{1} \neq p_{2}\). Sample data: \(x_{1}=804, n_{1}=874\) \(x_{2}=902, n_{2}=954\)

Short Answer

Expert verified
Fail to reject the null hypothesis. There is not enough evidence to conclude that \(p_{1} eq p_{2}\).

Step by step solution

01

- Formulate Hypotheses

Determine the null and alternative hypotheses. For this problem, the null hypothesis (H_{0}) and the alternative hypothesis (H_{a}) are: \(H_{0}: p_{1} = p_{2} \)\(H_{a}: p_{1} eq p_{2} \)
02

- Calculate Sample Proportions

Calculate the sample proportions for each group: \( \hat{p}_{1} = \frac{x_{1}}{n_{1}} = \frac{804}{874} \approx 0.920\)\(\hat{p}_{2} = \frac{x_{2}}{n_{2}} = \frac{902}{954} \approx 0.945\)
03

- Find the Combined Proportion

Calculate the combined sample proportion: \(\hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} = \frac{804 + 902}{874 + 954} = \frac{1706}{1828} \approx 0.934\)
04

- Compute the Test Statistic

Use the combined sample proportion to compute the test statistic (\(z\)-score).\[ z = \frac{\hat{p}_{1} - \hat{p}_{2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}} + \frac{1}{n_{2}})}} \]Substituting the values: \[ z = \frac{0.920 - 0.945}{\sqrt{0.934(1-0.934)(\frac{1}{874} + \frac{1}{954})}} \approx -1.40 \]
05

- Determine the Critical Value

Using the \(\alpha = 0.05\) level of significance, and since this is a two-tailed test, find the critical values for \(z\) from the standard normal distribution table. The critical values are \(z = \pm 1.96\)
06

- Calculate the P-value

Calculate the P-value corresponding to the test statistic \(|z| = 1.40\). Using the standard normal distribution table, find the P-value for \(z = 1.40\) which is approximately 0.1616 for each tail. Since this is a two-tailed test, multiply this by 2: \[ \text{P-value} \approx 2 \times 0.1616 = 0.3232 \]
07

- Make a Decision

Compare the P-value with the significance level \(\alpha = 0.05\). Since 0.3232 > 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that \(p_{1} eq p_{2} \)

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Key Concepts

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

null hypothesis
In hypothesis testing, the **null hypothesis** represents a statement of no effect or no difference. It is denoted as **H鈧�**. In this exercise, the null hypothesis is that the proportions from two different groups are equal. Formally, we write:
  • H鈧�: p鈧� = p鈧�
The null hypothesis serves as a starting point for testing. By assuming there is no difference, we can analyze if the sample data provides sufficient evidence to reject this claim. The goal is to determine whether the observed difference is due to random chance or if it's statistically significant.
alternative hypothesis
The **alternative hypothesis** is the statement we test against the null hypothesis. It represents what we aim to prove. Denoted as **H鈧�**, it reflects the presence of an effect or difference. In this example, the alternative hypothesis is that the proportions from two groups are not equal. Explicitly, it is:
  • H鈧�: p鈧� 鈮� p鈧�
Choosing the alternative hypothesis correctly is crucial because it defines the direction and nature of the test. Since our objective is to find if there's any difference, we use a two-tailed test, where deviations in both directions (higher or lower) will be considered.
test statistic
The **test statistic** quantifies the difference between the sample data and what is expected under the null hypothesis. Here, it is represented by the **z-score**, calculated using the combined proportions of the samples. The formula is:equation: \[ z = \frac{\hat{p_1} - \hat{p_2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]Using our exercise data, it becomes:
  • \(\hat{p_1} \approx 0.920\)
  • \(\hat{p_2} \approx 0.945\)
  • \(\hat{p} \approx 0.934\)
Substituting these values, we get:
  • \(z \approx -1.40\)
This z-score tells us how many standard deviations our sample proportions are from each other relative to the null hypothesis.
critical value
The **critical value** is a threshold we use to compare against the test statistic to determine whether to reject the null hypothesis. For a significance level (\(\alpha\)) of 0.05 in a two-tailed test, the critical values are from the standard normal distribution table:
  • \(z = \pm 1.96\) at \(\alpha = 0.05\)
If the test statistic falls beyond these critical values, we reject the null hypothesis. In our exercise, the critical values set the boundaries for decision-making. Because our computed z-score of **-1.40** is within this range (-1.96 < -1.40 < 1.96), we do not reject the null hypothesis.
P-value
The **P-value** provides the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It is a crucial tool for decision-making in hypothesis testing. For our z-score of \(|z| = 1.40\), we find the corresponding P-value from the standard normal distribution table:
  • P-value for **z = 1.40** is approximately 0.1616 per tail
Since it's a two-tailed test, we multiply by 2:
  • P-value = 2 * 0.1616 鈮� 0.3232
This P-value **(0.3232)** indicates the probability of observing such a result if the null hypothesis were true. Because it is greater than our significance level of 0.05, we fail to reject the null hypothesis, suggesting insufficient evidence to prove a difference.
proportion
The concept of **proportion** in statistics refers to part of the whole, representing a fraction of the population or sample. In this exercise, proportions are calculated by dividing the number of successful outcomes by the total number of trials in each sample. Notably:
  • For sample 1: \(\hat{p_1} = \frac{x_1}{n_1} = \frac{804}{874} \approx 0.920\)
  • For sample 2: \(\hat{p_2} = \frac{x_2}{n_2} = \frac{902}{954} \approx 0.945\)
These sample proportions describe the rates of success within each sample group. Additionally, the combined proportion (\(\hat{p}\)) aggregates data from both samples:
  • \(\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{1706}{1828} \approx 0.934\)
Understanding these proportions helps measure and compare different groups' characteristics, playing a central role in hypothesis testing.

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Most popular questions from this chapter

In a study published in the journal Teaching of Psychology, the article "Fudging the Numbers: Distributing Chocolate Influences Student Evaluations of an Undergraduate Course" states that distributing chocolate to students prior to teacher evaluations increases results. The authors randomly divided three sections of a course taught by the same instructor into two groups. Fifty of the students were given chocolate by an individual not associated with the course and 50 of the students were not given chocolate. The mean score from students who received chocolate was \(4.2,\) while the mean score for the nonchocolate groups was 3.9. Suppose that the sample standard deviation of both the chocolate and nonchocolate groups was 0.8 . Does chocolate appear to improve teacher evaluations? Use the \(\alpha=0.1\) level of significance.

Perform the appropriate hypothesis test. If \(n_{1}=31, s_{1}=12, n_{2}=51,\) and \(s_{2}=10,\) test whether \(\sigma_{1}>\sigma_{2}\) at the \(\alpha=0.05\) level of significance.

In October 1947, the Gallup organization surveyed 1100 adult Americans and asked, 鈥淎re you a total abstainer from, or do you on occasion consume, alcoholic beverages?鈥� Of the 1100 adults surveyed, 407 indicated that they were total abstainers. In a recent survey, the same question was asked of 1100 adult Americans and 333 indicated that they were total abstainers. Has the proportion of adult Americans who totally abstain from alcohol changed? Use the a = 0.05 level of significance.

Determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A political scientist wants to know how a random sample of 18- to 25-year-olds feel about Democrats and Republicans in Congress. She obtains a random sample of 1030 registered voters 18 to 25 years of age and asks, 鈥淒o you have favorable/unfavorable [rotated] opinion of the Democratic/Republican [rotated] party?鈥� Each individual was asked to disclose his or her opinion about each party.

Perform the appropriate hypothesis test. A random sample of \(n_{1}=135\) individuals results in \(x_{1}=40\) successes. An independent sample of \(n_{2}=150\) individuals results in \(x_{2}=60\) successes. Does this represent sufficient evidence to conclude that \(p_{1}
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