Question

A tourist agency in Florida claims the mean daily cost of means and lodging for a family of four traveling in Florida is $568. You work for a consumer protection advocate and want to test the claim. In a random sample of 50 families of four traveling in Florida, the mean daily cost of means and lodging is $584 and the standard deviation is $25

1. Do you have enough evidence to reject the agency’s claim?

2. What type of error is possible based on your conclusion?

3. What would be a consequence of this type of error in this situation?

Answer 1

Let be the true mean daily cost of meals and lodging for a family of four traveling in Florida. We want to test the claim that the mean daily cost of meals and lodging for a family of four traveling in Florida is $568. That is, we want to test the claim that

The hypotheses we want to test are

We will test the hypotheses at 5% level of significance

We have the following sample information

n=50 is the sample size

is the sample mean daily cost of meals and lodging for a family of four traveling in Florida.

is the sample standard deviation of daily cost of meals and lodging for a family of four traveling in Florida

We do not know the population standard deviation. We will estimate it using the sample

The estimated standard error of mean is

Since the sample size is greater than 30, using the central limit theorem, we can say that the sampling distribution of mean is normal. That is, we will use a 1 sample z test for mean.

The hypothesized value of mean is (from the null hypothesis)

The test statistic is

This is a 2 tailed test. (The alternative hypothesis has "not equal to").

The right tail critical value for is

Using the standard normal tables, we get for z=1.96, P(Z<1.96)=0.975.

The critical values are -1.96 and +1.96

We will reject the null hypothesis if the test statistic is not within the range of critical values.

Here, the test statistic is 4.525 and it is outside the range -1.96 and +1.96. Hence we reject the null hypothesis.

Do you have enough evidence to reject the agency’s claim?

ans: Yes: We have sufficient evidence to reject the agency's claim that the mean daily cost of meals and lodging for a family of four traveling in Florida is $568.

What type of error is possible based on your conclusion?

We have rejected the null hypothesis. That is we have rejected the agency's claim based on the sample evidence.  However, if the agency's claim were true, we would wrongly have rejected a true claim.

If we reject a the null hypothesis which is true, we commit a type I error.

ans: Type I error is possible, based on our conclusion as it is possible that we have rejected a null hypothesis, which is true.

What would be a consequence of this type of error in this situation?

ans: the consequence is, we would wrongly have rejected the true claim by the agency that the mean daily cost of meals and lodging for a family of four traveling in Florida is $568.