Dear Math Helper, You write a great column, and I hope you stay around for years to come. I run a small boarding school. We are
going to begin using test scores to help determine admissions eligibility. The scores for the test I have in mind are normally distributed with a mean of 300 and a standard deviation of 12.5. Can you show me how to determine the cutoff score if we should use to only accept only applicants who score in the top 10% on the test? What cutoff score includes the top 15%? Thanks in advance. Regards, The Student Selector
1 answer:
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Answer:
Since is a right tailed test the p value would be:
If we use a significance level of 0.05 we see that and then we can conclude that we don't have evidence in order to conclude that the mean is higher than 5.63, so then the claim makes sense.
Step-by-step explanation:
Data given and notation
represent the sample mean
represent the standard deviation for the sample
sample size
represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
represent the p value for the test (variable of interest)
State the null and alternative hypotheses to be tested
We need to conduct a hypothesis in order to determine if the average is more than 5.63, the system of hypothesis would be:
Null hypothesis:
Alternative hypothesis:
Compute the test statistic
We don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:
(1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
We can replace in formula (1) the info given like this:
Now we need to find the degrees of freedom for the t distribution given by:
P value
Since is a right tailed test the p value would be:
If we use a significance level of 0.05 we see that and then we can conclude that we don't have evidence in order to conclude that the mean is higher than 5.63, so then the claim makes sense.