Summary of Studies
Study
Sample
Size​ (N)
Population
Mean
Estimated
Population
Variance left parenthesis Upper S squared right parenthesisS2
Sample
Mean​ (M)
Tails
Significance
Level
​(a)
6464
12.6412.64
8.458.45
10.7110.71
1​ (low predicted)
0.05
​(b)
4949
1 comma 007.011,007.01
319.09319.09
1 comma 009.431,009.43
2
0.01
​(c)
400400
51.6051.60
6.656.65
52.6252.62
1​ (high predicted)
0.01
Decide whether the result from Study​ (a) is significant. Let the first population be the one from which the sample is taken and let the second population be the one whose mean is given. What is the null hypothesis for this​ test?
A.
The mean for the first population is not different from the mean of the second population.
Your answer is correct.B.
The mean for the first population is different from the mean of the second population.
C.
The mean for the first population is less than the mean of the second population.
D.
The mean for the first population is greater than the mean of the second population.
Part 2
What is the research hypothesis for this​ test?
A.
The mean for the first population is different from the mean of the second population.
B.
The mean for the first population is less than the mean of the second population.
Your answer is correct.C.
The mean for the first population is not different from the mean of the second population.
D.
The mean for the first population is greater than the mean of the second population.
Part 3
Determine the cutoff sample​ score(s) on the comparison distribution at which the null hypothesis should be rejected.
The cutoff sample​ score(s) is(are) enter your response here.
​(Use a comma to separate answers as needed. Round to three decimal places as​ needed.)

### Summary of Studies:
Here is a summary of the provided data for the three studies:

| **Study** | **Sample Size (N)** | **Population Mean** | **Estimated Population Variance (S²)** | **Sample Mean (M)** | **Tails** | **Significance Level (α)** |
|----------|---------------------|---------------------|-----------------------------------------|---------------------|-----------|----------------------------|
| (a)      | 64                   | 12.64               | 8.45                                    | 10.71               | 1 (low predicted) | 0.05                       |
| (b)      | 49                   | 1,007.01            | 319.09                                  | 1,009.43            | 2         | 0.01                       |
| (c)      | 400                  | 51.60               | 6.65                                    | 52.62               | 1 (high predicted) | 0.01                       |

### Part 1: Null Hypothesis for Study (a)
The null hypothesis (\(H_0\)) tests whether the sample mean differs significantly from the population mean. For Study (a), the null hypothesis is:
- **A**: *The mean for the first population is not different from the mean of the second population.*

This option is correct. The null hypothesis always assumes no difference between the population and sample means.

### Part 2: Research Hypothesis for Study (a)
The research hypothesis (\(H_1\)) contrasts the null hypothesis and usually reflects what the researcher expects. Given that Study (a) is one-tailed (low predicted), the research hypothesis is:
- **B**: *The mean for the first population is less than the mean of the second population.*

This option is correct since it's a one-tailed test where the predicted direction is "lower."

### Part 3: Cutoff Sample Score for Study (a)
To determine the cutoff sample score, we calculate the critical z-value for a one-tailed test with a significance level \( \alpha = 0.05 \).

1. **Significance Level**: \( \alpha = 0.05 \) for a one-tailed test.
2. **Z-critical value**: The z-value corresponding to \( \alpha = 0.05 \) is:
   - For a one-tailed test, the critical z-value is \( -1.645 \).

Thus, the cutoff sample score is \( -1.645 \).

### Answer: The cutoff sample score(s) is/are **-1.645**.

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Let me know if you need any further details or clarifications!

### Related Questions:
1. How do we calculate the test statistic using the sample mean and population mean?
2. What are the conditions for rejecting the null hypothesis in a one-tailed test?
3. How does changing the significance level (\( \alpha \)) affect the cutoff score?
4. What is the importance of the population variance in hypothesis testing?
5. How would the cutoff score change if the test were two-tailed instead?

### Tip:
In hypothesis testing, the significance level (\( \alpha \)) represents the probability of rejecting a true null hypothesis (Type I error). Lowering \( \alpha \) reduces this risk but makes it harder to detect real differences.

Decide whether the result from Study (a) is significant. Let the first population be the one from which the sample is taken and let the second population be the one whose mean is given. What is the null hypothesis for this test? A. The mean for the first population is not different from the mean of the second population. B. The mean for the first population is different from the mean of the second population. C. The mean for the first population is less than the mean of the second population. D. The mean for the first population is greater than the mean of the second population.

Understand how to determine the significance of results from Study (a) using hypothesis testing. This problem covers concepts like null and alternative hypotheses, significance levels, and z-scores, with a one-tailed test at α = 0.05. Suitable for college-level statistics students.

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