A group of 200 students takes the Kaplan course in hopes that it will improve their SAT scores above the national mean of 500 on the math section. The population standard deviation of the SAT is known to be 100. Use an alpha level of .05 when answering the following questions. Compute the one-sample z-test statistic.
To compute the one-sample z-test statistic, you need to follow these steps:
Step 1: State the null hypothesis and alternative hypothesis.
The null hypothesis, denoted as H0, is a statement that assumes there is no significant difference between the sample mean and the population mean. In this case, the null hypothesis is that the Kaplan course does not improve the students' SAT scores above the national mean of 500 on the math section:
H0: μ = 500
The alternative hypothesis, denoted as Ha, is a statement that assumes there is a significant difference between the sample mean and the population mean. In this case, the alternative hypothesis is that the Kaplan course does improve the students' SAT scores above the national mean of 500 on the math section:
Ha: μ > 500
Step 2: Determine the level of significance (alpha).
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is true. The given alpha level is 0.05, meaning that we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Calculate the test statistic.
The formula to calculate the one-sample z-test statistic is given by:
z = (X - μ) / (σ / √n)
Where:
X = sample mean (mean SAT score of the 200 students)
μ = population mean (national mean SAT score of 500)
σ = population standard deviation (known to be 100)
n = sample size (number of students, which is 200 in this case)
Step 4: Plug in the values and calculate the test statistic.
z = (X - μ) / (σ / √n)
z = (X - 500) / (100 / √200)
Now, you need to know the sample mean (X) to calculate the test statistic. Once you have that value, plug it into the formula above, along with the other known values, to calculate the one-sample z-test statistic.
Formula for one-sample z-test statistic:
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)
I'll let you take it from here.