We obtain a X = 46.8 (with N = 15). This sample may represent the population where u = 50 (o, = 11). Using the .05 criterion and the lower tail of the sampling distribution: (a) What is our critical value? (b) Is this sample in the region of rejection? How do you know? (c) What should we conclude about the sample and why?
We obtain a X = 46.8 (with N = 15). This sample may represent the population where u = 50 (o, = 11). Using the .05 criterion and the lower tail of the sampling distribution: (a) What is our critical value? (b) Is this sample in the region of rejection? How do you know? (c) What should we conclude about the sample and why?
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.3: Measures Of Spread
Problem 26PFA
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![**Problem Statement:**
We obtain a sample mean (\(\overline{X}\)) of 46.8 with a sample size (N) of 15. This sample may represent the population where the population mean (μ) is 50 and the population standard deviation (σ_X) is 11. Using a significance level (α) of .05 and considering the lower tail of the sampling distribution:
**Questions:**
(a) What is our critical value?
(b) Is this sample in the region of rejection? How do you know?
(c) What should we conclude about the sample and why?
**Solution Steps:**
1. **Determine the Critical Value:**
For a one-tailed test with α = .05, find the critical value using the Z-table. The critical value (Z-critical) is the Z-score corresponding to the cumulative probability of 0.05 in the lower tail.
2. **Compare the Sample Mean with Critical Value:**
Calculate the Z-score for the sample mean using the formula:
\[
Z = \frac{\overline{X} - \mu}{\sigma/\sqrt{N}}
\]
Compare this Z-score with the critical value obtained in step 1.
3. **Conclusion About the Sample:**
Determine whether the sample mean falls in the region of rejection (Z < Z-critical). Conclude whether the sample mean is significantly different from the population mean based on this comparison.
Each of the calculations and steps should be explained in detail on the educational website, along with the background information on hypothesis testing and the Z-table.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31c50aa7-6739-4a15-b164-b3419a9ef375%2F60f62410-2398-4099-aae5-8281b1736f39%2Fx689ael_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
We obtain a sample mean (\(\overline{X}\)) of 46.8 with a sample size (N) of 15. This sample may represent the population where the population mean (μ) is 50 and the population standard deviation (σ_X) is 11. Using a significance level (α) of .05 and considering the lower tail of the sampling distribution:
**Questions:**
(a) What is our critical value?
(b) Is this sample in the region of rejection? How do you know?
(c) What should we conclude about the sample and why?
**Solution Steps:**
1. **Determine the Critical Value:**
For a one-tailed test with α = .05, find the critical value using the Z-table. The critical value (Z-critical) is the Z-score corresponding to the cumulative probability of 0.05 in the lower tail.
2. **Compare the Sample Mean with Critical Value:**
Calculate the Z-score for the sample mean using the formula:
\[
Z = \frac{\overline{X} - \mu}{\sigma/\sqrt{N}}
\]
Compare this Z-score with the critical value obtained in step 1.
3. **Conclusion About the Sample:**
Determine whether the sample mean falls in the region of rejection (Z < Z-critical). Conclude whether the sample mean is significantly different from the population mean based on this comparison.
Each of the calculations and steps should be explained in detail on the educational website, along with the background information on hypothesis testing and the Z-table.
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