Although there is a popular belief that herbal remedies such as Ginkgo biloba and Ginseng may improve learning and memory in healthy adults, these effects are usually not supported by well-controlled research (Persson, Bringlov, Nilsson, and Nyberg, 2004). In a study, a researcher obtains a sample of n = 16 participants and has each person take the herbal supplements every day for 90 days. At the end of the 90 days, each person takes a standardized memory test. For the general population, scores from the test form a normal distribution with a mean of μ = 50 and a standard deviation of σ = 12. The sample of research participants had an average of M = 54. Using the standard 4-step procedure, conduct a two-tailed hypothesis test with a .05 level of significance, α = .05, and p = .059 to evaluate the effect of the supplements.

o Step 1: State the hypotheses. State the null hypothesis and the alternative hypothesis. Write the hypotheses using statistical notation and for each, include a statement of interpretation in plain English.
o Step 2: Set the criteria for a decision. State the level of significance, the alpha level, and whether the test is one-tailed (directional) or two-tailed (nondirectional).
o Step 3: Compute the test statistic. State the observed p value.
o Step 4: Make a decision. State whether to retain or reject the null hypothesis.

We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.

I will start you out. State with your own notations.

Ho: no difference
H1: difference

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score to compare to your level of significance.

Step 1: State the hypotheses.

Null hypothesis (H0) - There is no effect of the herbal supplements on memory performance in healthy adults.
Alternative hypothesis (HA) - The herbal supplements have an effect on memory performance in healthy adults.

Plain English interpretation: The null hypothesis states that taking herbal supplements does not improve memory in healthy adults, while the alternative hypothesis suggests that taking herbal supplements does have an effect on memory in healthy adults.

Step 2: Set the criteria for a decision.

Level of significance (α) = 0.05 (or 5%)
The test is two-tailed (nondirectional) because we are interested in whether there is any effect of the supplements, regardless of the direction.

Step 3: Compute the test statistic.

From the information given, we know that the sample mean (M) is 54, the population mean (μ) is 50, and the population standard deviation (σ) is 12. We also know that the sample size (n) is 16.

We can calculate the standard error of the mean (SEM) using the formula: SEM = σ / sqrt(n)
SEM = 12 / sqrt(16) = 3

We can then calculate the z-score using the formula: z = (M - μ) / SEM
z = (54 - 50) / 3 = 4 / 3 = 1.33

Step 4: Make a decision.

The observed p-value is given as p = 0.059.

Since the p-value (0.059) is greater than the alpha level (0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the herbal supplements have a significant effect on memory performance in healthy adults.

Step 1: State the hypotheses

The null hypothesis (H0): The herbal supplements have no effect on memory performance in healthy adults.
The alternative hypothesis (Ha): The herbal supplements have an effect on memory performance in healthy adults.

Plain English interpretation:
H0: The average memory scores of participants who took the herbal supplements for 90 days are not different from the general population mean.
Ha: The average memory scores of participants who took the herbal supplements for 90 days are different from the general population mean.

Step 2: Set the criteria for a decision

Level of significance (α): 0.05 (or 5%)
Type of test: Two-tailed (nondirectional) test

Step 3: Compute the test statistic

To compute the test statistic, we need to calculate the z-score using the sample mean, population mean, and standard deviation.

Z = (sample mean - population mean) / (standard deviation / square root of the sample size)

In this case:
Sample mean (M) = 54
Population mean (μ) = 50
Standard deviation (σ) = 12
Sample size (n) = 16

Z = (54 - 50) / (12 / sqrt(16))
Z = 4 / (12 / 4)
Z = 4 / 3
Z = 1.333

To find the observed p-value, we can refer to a standard normal distribution table or use statistical software.

The observed p-value is 0.183.

Step 4: Make a decision

Since this is a two-tailed test, we will compare the observed p-value with the level of significance (α).

If the observed p-value is less than α/2 (0.05/2 = 0.025) or greater than 1 - α/2 (1 - 0.05/2 = 0.975), we reject the null hypothesis.
If the observed p-value is between α/2 and 1 - α/2, we fail to reject the null hypothesis.

In this case, the observed p-value (0.183) is greater than 0.025 but less than 0.975.

Therefore, we fail to reject the null hypothesis.

Conclusion:
The results of the hypothesis test indicate that there is not enough evidence to conclude that the herbal supplements have a significant effect on memory performance in healthy adults.