The mean of math SAT scores is 500 and the standard deviation is 100. A random sample of 64 students from Leeds H.S. received an average of 480. Conduct a hypothesis test to determine if the population mean SAT score for Leeds H.S. is less than 500 with alpha = .05. Determine a p-value and explain it.
I know I need a one-tail test.
Yes, you are correct. This will be a one-tailed test (the alternate hypothesis shows a specific direction on a one-tailed test). Use a z-test. See your other post for an example on calculating the z-test statistic using a z-test.
Your hypotheses for this problem:
Ho: µ = 500 -->null hypothesis
Ha: µ < 500 -->alternate hypothesis
The p-value is the actual level of the test statistic which you can find using a z-table. Compare with .05 to determine whether or not to reject the null. If you reject the null, then you can conclude µ < 500.
I hope this will help get you started.
To conduct a hypothesis test, we need to set up the null and alternate hypotheses and calculate the test statistic and p-value.
Null hypothesis (H0): The population mean SAT score for Leeds H.S. is equal to 500.
Alternate hypothesis (Ha): The population mean SAT score for Leeds H.S. is less than 500.
Since we want to test if the mean SAT score is less than 500, it is a one-tailed test. The significance level (alpha) is given as 0.05.
To calculate the test statistic, we can use the formula:
test statistic = (sample mean - population mean) / (standard deviation / sqrt(sample size))
In this case, the sample mean is 480, the population mean is 500, the standard deviation is 100, and the sample size is 64. Plugging in the values, we get:
test statistic = (480 - 500) / (100 / sqrt(64))
= -20 / (100 / 8)
= -20 / 12.5
= -1.6
To find the p-value, we need to determine the probability of observing a test statistic of -1.6 or less under the null hypothesis. We can use a Z-table or statistical software to find the p-value associated with the test statistic.
Using a standard normal distribution table or a stats calculator, we can find that the p-value associated with a test statistic of -1.6 is approximately 0.0548.
p-value = 0.0548
Since the p-value (0.0548) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the population mean SAT score for Leeds H.S. is less than 500 at a 5% level of significance.