A Researcher Is Testing The Claim That Adults Consume An Ave
A Researcher Is Testing The Claim That Adults Consume An Average Of
A researcher is testing the claim that adults consume an average of at least 1.85 cups of coffee per day. A sample of 35 adults shows a sample mean of 1.75 cups per day with a sample standard deviation of 0.4 cups per day. Test the claim at a 5% level of significance. What is your conclusion?
Report the p-value for this test.
A government bureau claims that more than 50% of U.S. tax returns were filed electronically last year. A random sample of 150 tax returns for last year contained 80 that were filed electronically. Test the bureau's claim at a 5% level of significance. What is your conclusion?
Report the p-value for this test.
A major automobile company claims that its new electric-powered car has an average range of more than 100 miles. A random sample of 40 new electric cars was selected to test the claim. Assume that the population standard deviation is 12 miles. A 5% level of significance will be used for the test.
A) What would be the consequences of making a Type II error in this problem?
B) Compute the probability of making a Type II error if the true population mean is 105 miles.
C) What is the maximum probability of making a Type I error in this problem?
Please note: A hypothesis test answer must contain: a null and an alternative hypothesis, a computed value of the test statistic, a critical value of the test statistic, a decision, and a conclusion.
Paper For Above instruction
Introduction
Statistical hypothesis testing is an essential component of empirical research, allowing researchers to make informed decisions about population parameters based on sample data. This paper addresses three specific hypothesis testing scenarios involving population means and proportions, emphasizing the application of significance levels, p-values, and errors in decision-making processes. The three cases involve testing the average daily coffee consumption among adults, the proportion of electronic tax returns, and the average range of a new electric vehicle.
Case 1: Adult Coffee Consumption
The first hypothesis test concerns whether adults consume at least 1.85 cups of coffee daily on average. The null hypothesis (H₀) states that the true mean is at least 1.85, while the alternative hypothesis (H₁) suggests the mean is less than 1.85. Formally:
H₀: μ ≥ 1.85
H₁: μ
Given the sample data—a sample mean (\(\bar{x}\)) of 1.75, a sample size (n) of 35, and a sample standard deviation (s) of 0.4—this is a t-test for a mean with an unknown population standard deviation. The calculated t-statistic is:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{1.75 - 1.85}{0.4 / \sqrt{35}} \approx -1.84 \]
The critical value at \(\alpha = 0.05\) for a one-tailed t-test with 34 degrees of freedom is approximately -1.690. Since the calculated t is -1.84
The p-value associated with t = -1.84 and df=34 is approximately 0.037, which is less than 0.05, reinforcing the decision to reject H₀. Therefore, the evidence suggests that adults consume less than 1.85 cups on average.
Case 2: Proportion of Electronically Filed Tax Returns
The second case tests if more than 50% of tax returns were filed electronically. Here, the sample proportion (\(\hat{p}\)) is 80/150 ≈ 0.533. The hypotheses are:
H₀: p ≤ 0.50
H₁: p > 0.50
Using a z-test for proportions, the test statistic is:
\[ z = \frac{\hat{p} - p_0}{\sqrt{p_0 (1 - p_0) / n}} = \frac{0.533 - 0.50}{\sqrt{0.50 \times 0.50 / 150}} \approx 1.54 \]
The critical value at \(\alpha = 0.05\) for a one-tailed z-test is approximately 1.645. Since 1.54
The p-value corresponding to z = 1.54 is approximately 0.061. Because this p-value exceeds 0.05, this aligns with the failure to reject H₀.
Case 3: Electric Car Range
In this scenario, the claim is that the average range exceeds 100 miles, with the true population mean potentially being 105 miles. The hypotheses:
H₀: μ ≤ 100
H₁: μ > 100
Given: sample size (n) = 40, sample mean (\(\bar{x}\)) is unknown, but for the purpose of the question (calculating Type II error at μ = 105), the population standard deviation (σ) = 12 miles, and significance level \(\alpha = 0.05\).
Suppose the sample mean corresponds to the true mean of 105 miles. The test statistic:
\[ z = \frac{\bar{x} - 100}{\sigma / \sqrt{n}} \]
To compute the probability of a Type II error (β) when the true mean is 105, find the critical z-value for the rejection region:
\[ z_{critical} = z_{1-\alpha} = 1.645 \]
Rearranged to find the critical sample mean:
\[ \bar{x}_{critical} = 100 + z_{critical} \times \frac{\sigma}{\sqrt{n}} = 100 + 1.645 \times \frac{12}{\sqrt{40}} \approx 104.24 \text{ miles} \]
The probability of failing to reject H₀ when μ = 105 involves calculating the Z-score:
\[ Z = \frac{\bar{x} - \mu_{true}}{\sigma / \sqrt{n}} \]
At the critical point:
\[ Z = \frac{104.24 - 105}{12 / \sqrt{40}} \approx -0.478 \]
The probability (β) that the test statistic falls below the critical value when the true mean is 105 is:
\[ \beta \approx P(Z
So, there's approximately a 31.7% chance of Type II error if the true mean is 105 miles.
Finally, the maximum probability of a Type I error (α) in this test, as specified, is 5% (0.05).
Discussion on Errors and Implications
A Type II error occurs when a false null hypothesis is not rejected. In the context of the electric vehicle range, failing to reject H₀ when the true mean is 105 miles would suggest that the company or regulators might wrongly assume the vehicle's range does not exceed 100 miles, potentially impacting consumer expectations or regulatory approvals. Making this error can lead to underestimating the vehicle's performance and may influence market competitiveness and consumer trust negatively.
Conversely, a Type I error involves rejecting a true null hypothesis. In this case, incorrectly concluding that the range exceeds 100 miles when it does not might lead the company to overstate the vehicle's capabilities, risking reputation and compliance issues if the vehicle does not meet the claimed range in real-world conditions.
The calculated probability of a Type II error (β ≈ 31.7%) indicates a moderate risk of missing a true effect, emphasizing the importance of selecting an appropriate sample size for reliable testing. To reduce this risk, increasing the sample size or adjusting significance levels could improve test sensitivity.
Conclusion
In hypothesis testing, understanding the balance between Type I and Type II errors is crucial. The first case demonstrates a significant result suggesting that adults consume less than 1.85 cups of coffee daily. The second case shows insufficient evidence to claim that more than half of the tax returns are filed electronically. The third case emphasizes the importance of sample size and error probability considerations in evaluating vehicle performance claims. Overall, effective hypothesis testing involves careful formulation of hypotheses, calculation of test statistics, interpretation of p-values, and awareness of potential errors and their consequences.
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