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.