A study found that for adults the general self-efficacy scale mean score was 35 and standard deviation was 12. A group of clients (n=14) was measured for their general self-efficacy with the same measure and found having 40 with standard deviation of 20. A group of therapist would like to know if the group self-efficacy mean score is significantly different from the population mean score use p=0.05.
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 related to the Z score.
To determine if the group self-efficacy mean score is significantly different from the population mean score, you can perform a hypothesis test using the z-test.
Here's how you can calculate it step by step:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The group self-efficacy mean score is equal to the population mean score (μ = 35).
- Alternative hypothesis (HA): The group self-efficacy mean score is significantly different from the population mean score (μ ≠ 35).
Step 2: Set the level of significance (α):
In this case, the significance level is given as p = 0.05.
Step 3: Calculate the test statistic (z-score):
The formula to calculate the z-score is:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case:
x = 40 (sample mean)
μ = 35 (population mean)
σ = 12 (population standard deviation)
n = 14 (sample size)
Substituting the values into the formula:
z = (40 - 35) / (12 / sqrt(14))
Step 4: Determine the critical value:
Since the alternative hypothesis is two-tailed (μ ≠ 35), we need to find the critical value(s) for a two-tailed test at a 95% confidence level.
Using a z-table or a statistical calculator, the critical values for a two-tailed test at α = 0.05 are approximately ±1.96.
Step 5: Compare the test statistic with the critical value:
If the absolute value of the test statistic is greater than the critical value (1.96), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, if |z| > 1.96, we reject the null hypothesis (H0). If |z| ≤ 1.96, we fail to reject the null hypothesis (H0).
Step 6: Calculate the p-value (optional):
If you are interested in calculating the p-value, you can use the z-score to find the probability associated with it.
For a two-tailed test, you need to find the area in both tails of the distribution that is more extreme than the test statistic. Multiply this area by 2 to get the p-value.
Step 7: Make a conclusion:
If the p-value is less than the significance level (α), typically 0.05, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Note: The p-value is not necessary if you are only interested in determining if the mean score is significantly different (as stated in this question) and not specifically interested in the exact probability.
By following the above steps, you can determine if the group self-efficacy mean score is significantly different from the population mean score.