Hypothesis Test

[NnmB35]

I am trying to understand the hypothesis test - specifically, which side of the normal curve the rejection region should lie and what my answer means.
Example:
Restaurant claims that mean wait time is 3 minutes with sd = 1 minute. Sample of 50 customers found mean = 2.75 minutes. At 0.05 significance, can we conclude that mean wait time is less than 3 minutes?
Here is what I did:
Ho: u <= 3; H1: u >3.
Critical value = 1.65 (0.5 - 0.05 = 0.45, z for 0.45)
Z = Xbar - u/(s/ SQRT(n))
= 2.75 - 3/(1/SQRT50)
= -1.77
At a Z of 1.77, my % is 0.46.
Am I correct so far?

To interpret - the likelihood of finding a z value of -1.77 or greater when Ho is true is 0.96 (0 to -1.77, + 0.5 from the right side of the curve). Therefore the Ho is true...
Is this the correct interpretation of the data?

Thank you

 

[BioStatMatt]

You said:

Restaurant claims that mean wait time is 3 minutes with sd = 1 minute. Sample of 50 customers found mean = 2.75 minutes. At 0.05 significance, can we conclude that mean wait time is less than 3 minutes?

In general, the null hypothesis is does not include a "greater than" or "less than" statement, rather only an "equals" statement. For example:

Ho: mu = 0

Typically the alternative hypothesis for the mean will include a "greater than", "less than", or "not equal to" statement:

H1: mu < 0

In your case we will set the null hypothesis to what the restaurant claims to be true:

Ho: mu = 3

But since our sample suggested that the mean may be less than that, we would like to test

H1: mu < 3

Next we would like to approximate this problem with the standard normal and define a critical region such that we are 95% confident that the true mean (mu) does not lie within this region.

The critical value for this test is -1.645, that is 95% of the observations from the standard normal distribution will lie above this value. So the critical region is the set of values below -1.645. We know that if our test statistic (Z) is within the critical region then we reject the null hypothesis.

Next is to transform our sample mean to a standard normal random varaible by

Z = Xbar - u/(s/ SQRT(n))

If you made your calculations correctly, -1.77 is within the critical region, and we reject the null hypothesis.

~Matt