For each of the following samples that were given an experimental treatment, test whether they represent populations that score significantly higher than the general population: (a) a smaple of 100 with a mean of 82, (b) a sample of 10 with a mean of 84. The general population of individuals has a mean of 81, a standard deviation of 8, and follows a normal curve. For each sample, carry out a Z test using the five steps of hypothesis testing with a one-tailed test at the .01 significance level, and make a drawing of the distributions involved. (c) Advanced topic: Figure the 99% confidence interval for parts (a) and (b).
To test whether the given samples represent populations that score significantly higher than the general population, we can follow the five steps of hypothesis testing and carry out a Z test. Here's how you can approach this:
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H0): The sample does not represent a population that scores significantly higher than the general population.
- Alternative Hypothesis (Ha): The sample represents a population that scores significantly higher than the general population.
Step 2: Determine the significance level.
In this case, the significance level is given as .01, which means we need to test at the 99% confidence level. The critical value for a one-tailed test at this level is approximately 2.33.
Step 3: Calculate the test statistic.
To calculate the Z test statistic, we need to use the formula: Z = (sample mean - population mean) / (population standard deviation / square root of sample size)
For sample (a):
Z = (82 - 81) / (8 / sqrt(100))
Z = 1 / (8 / 10)
Z = 1.25
For sample (b):
Z = (84 - 81) / (8 / sqrt(10))
Z = 3 / (8 / sqrt(10))
Z ≈ 1.89
Step 4: Determine the critical region.
Since we are performing a one-tailed test with the alternative hypothesis stating that the sample mean is significantly higher, we will only look at the upper tail of the distribution. The critical region is any Z value greater than the critical value of 2.33.
Step 5: Make a decision and interpretation.
For sample (a), the calculated Z value (1.25) falls short of the critical value (2.33). Therefore, we fail to reject the null hypothesis. This indicates that there is insufficient evidence to conclude that the sample represents a population that scores significantly higher than the general population.
For sample (b), the calculated Z value (1.89) exceeds the critical value (2.33). Thus, we reject the null hypothesis. This suggests that there is enough evidence to conclude that the sample represents a population that scores significantly higher than the general population.
Regarding the 99% confidence interval for parts (a) and (b), we can calculate it using the formula: Confidence Interval = sample mean ± (critical value * standard error)
For sample (a):
Standard Error = population standard deviation / sqrt(sample size)
Standard Error = 8 / sqrt(100) ≈ 0.8
Critical Value for 99% confidence = 2.33 (from earlier)
Confidence Interval = 82 ± (2.33 * 0.8) = (80.46, 83.54)
For sample (b):
Standard Error = 8 / sqrt(10) ≈ 2.53
Critical Value for 99% confidence = 2.33 (same as above)
Confidence Interval = 84 ± (2.33 * 2.53) = (78.06, 89.94)
By using the above calculations, we have obtained the results for both the hypothesis testing and the confidence intervals for samples (a) and (b).