Problem 21
Question
The following hypotheses are given. $$\begin{aligned}H_{0}: & \pi \leq .70 \\\H_{1}: & \pi>.70\end{aligned}$$ A sample of 100 observations revealed that \(p=.75 .\) At the .05 significance level, can the null hypothesis be rejected? a. State the decision rule. b. Compute the value of the test statistic. c. What is your decision regarding the null hypothesis?
Step-by-Step Solution
Verified Answer
Do not reject the null hypothesis; there is not enough evidence to claim that \( \pi > 0.70 \) at the 0.05 level.
1Step 1: Understand the Hypotheses
The null hypothesis is stated as \( H_0: \pi \leq 0.70 \) and the alternative hypothesis is \( H_1: \pi > 0.70 \). This indicates a one-tailed test to see if the true population proportion \( \pi \) is greater than 0.70.
2Step 2: State the Decision Rule
Since we are testing a proportion, we use the normal approximation for the test statistic. The decision rule is: reject \( H_0 \) if the test statistic is greater than the critical value from a standard normal distribution at the \( \alpha = 0.05 \) level, which is approximately 1.645 for a one-tailed test.
3Step 3: Calculate the Test Statistic
The test statistic for proportions is calculated using the formula:\[Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\]where \( \hat{p} = 0.75 \) is the sample proportion, \( p_0 = 0.70 \) is the null hypothesis proportion, and \( n = 100 \) is the sample size. Plugging in the values:\[Z = \frac{0.75 - 0.70}{\sqrt{\frac{0.70 \times 0.30}{100}}} = \frac{0.05}{0.0458} \approx 1.0915\]
4Step 4: Make the Decision
Compare the calculated test statistic (1.0915) with the critical value (1.645). Since 1.0915 is less than 1.645, we do not reject the null hypothesis \( H_0 \).
5Step 5: Conclusion: Decision on the Null Hypothesis
The test statistic does not exceed the critical value at the 0.05 significance level; therefore, we do not have enough evidence to reject the null hypothesis. We conclude that there is insufficient evidence to support the claim that the true population proportion is greater than 0.70.
Key Concepts
Population ProportionTest StatisticNull HypothesisDecision Rule
Population Proportion
In hypothesis testing, the term "population proportion" is essential when dealing with categorical data. It represents the ratio of individuals in a population showing a particular attribute relative to the total population. For example, if we wanted to know the proportion of people who prefer chocolate ice cream over vanilla, this proportion would be derived from survey responses.
In statistical symbols, we denote the population proportion by \(\pi\). It's the proportion we often speculate about, like in our exercise where we examined if \(\pi > 0.70\). The observed sample proportion is represented by \(\hat{p}\), which is found by dividing the number of successful outcomes by the total number of observations.
When performing tests, we take samples because evaluating the entire population is not always feasible. Thus, the sample proportion \(\hat{p}\) helps to estimate \(\pi\). However, remember that \(\hat{p}\) may vary from sample to sample, and it's important to consider this variability when making decision rules.
In statistical symbols, we denote the population proportion by \(\pi\). It's the proportion we often speculate about, like in our exercise where we examined if \(\pi > 0.70\). The observed sample proportion is represented by \(\hat{p}\), which is found by dividing the number of successful outcomes by the total number of observations.
When performing tests, we take samples because evaluating the entire population is not always feasible. Thus, the sample proportion \(\hat{p}\) helps to estimate \(\pi\). However, remember that \(\hat{p}\) may vary from sample to sample, and it's important to consider this variability when making decision rules.
Test Statistic
The test statistic is at the heart of hypothesis testing, functioning as the bridge between your sample data and the decision you make regarding the hypothesis.
In the context of population proportions, the test statistic commonly used is the \(Z\)-statistic. It quantifies how far the observed sample proportion \(\hat{p}\) is from the hypothesized population proportion \(p_0\), in units of standard error, formulated as:
Plugging these into the formula, the computed \(Z\) value turns out to be approximately 1.0915. This value, when compared to the critical value determines if the difference between sample and population proportion is statistically significant under the assumed distribution.
In the context of population proportions, the test statistic commonly used is the \(Z\)-statistic. It quantifies how far the observed sample proportion \(\hat{p}\) is from the hypothesized population proportion \(p_0\), in units of standard error, formulated as:
- \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\)
Plugging these into the formula, the computed \(Z\) value turns out to be approximately 1.0915. This value, when compared to the critical value determines if the difference between sample and population proportion is statistically significant under the assumed distribution.
Null Hypothesis
The null hypothesis, one of the fundamental pillars of hypothesis testing, represents a default position or a statement that there is no effect or no difference. In simpler terms, it is the assumption of "no change" which we aim to test using our sample data.
In our specific exercise, the null hypothesis \( H_0: \pi \leq 0.70 \) proposes that the true population proportion isn't greater than 0.70. Essentially, it's a starting point of our inquiry until the sample evidence suggests otherwise.
Rejecting the null implies that the sample data provides enough evidence to support the alternative hypothesis (here, \( H_1: \pi > 0.70 \)). If the test statistic falls within the critical region—determined by our level of significance and the distribution—it leads to concluding that the null hypothesis may not hold true for the population. However, if we don't find strong evidence against the null hypothesis, we "fail to reject" it, as in our example.
In our specific exercise, the null hypothesis \( H_0: \pi \leq 0.70 \) proposes that the true population proportion isn't greater than 0.70. Essentially, it's a starting point of our inquiry until the sample evidence suggests otherwise.
Rejecting the null implies that the sample data provides enough evidence to support the alternative hypothesis (here, \( H_1: \pi > 0.70 \)). If the test statistic falls within the critical region—determined by our level of significance and the distribution—it leads to concluding that the null hypothesis may not hold true for the population. However, if we don't find strong evidence against the null hypothesis, we "fail to reject" it, as in our example.
Decision Rule
A decision rule in hypothesis testing provides a guideline for deciding whether to reject the null hypothesis, based on the value of the test statistic. The rule is framed using a critical value derived from the statistical distribution relevant to the test.
For one-tailed tests involving population proportions, as in our exercise, the normal distribution is used. With an \(\alpha = 0.05\) significance level in a right-tailed test, the critical value is approximately 1.645. The decision rule then is:
For one-tailed tests involving population proportions, as in our exercise, the normal distribution is used. With an \(\alpha = 0.05\) significance level in a right-tailed test, the critical value is approximately 1.645. The decision rule then is:
- Reject \(H_0\) if the \(Z\)-statistic \( > 1.645\)
- Fail to reject \(H_0\) if the \(Z\)-statistic \( \leq 1.645\)
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