A rental agent claims that the mean monthly rent, u , for apartments on the east side of town is less than $675. A random sample of 21 monthly rents for apartments on the east side has a mean of $674, with a standard deviation of $18. If we assume that the monthly rents for apartments on the east side are normally distributed, is there enough evidence to conclude, at the 0.1 level of significance, that u is less than $675?

Perform a one-tailed test.
null hypothesis:
alternative hypothesis?
type of test statistic?
Critical value at the 0.1 level of significance?
Using the 0.1 level of significance, can we conclude that the mean monthly rent for apartments on the east side is less that $675?

To determine whether there is enough evidence to conclude that the mean monthly rent for apartments on the east side is less than $675, we need to perform a hypothesis test. Here are the steps:

1. Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The mean monthly rent for apartments on the east side is $675 or more (u ≥ $675)
- Alternative hypothesis (Ha): The mean monthly rent for apartments on the east side is less than $675 (u < $675)

2. Determine the type of test statistic appropriate for this situation:
Since we have a sample mean and the population standard deviation is known, we will use the z-test.

3. Find the critical value for the given level of significance:
The level of significance is 0.1 (or 10%), which means we want to reject the null hypothesis if the test statistic is less than our critical value. Referencing the z-table or a calculator, the critical value for a one-tailed test at the 0.1 level of significance is approximately -1.28.

4. Calculate the test statistic:
The test statistic, also known as the z-score, can be calculated using the formula:
z = (sample mean - hypothesized mean) / (standard deviation / square root of sample size)
In this case, the sample mean (x̄) is $674, hypothesized mean (μ) is $675, standard deviation (σ) is $18, and sample size (n) is 21.

z = ($674 - $675) / ($18 / √21)
= (-1) / (2.086)
≈ -0.479

5. Make a decision and interpret the results:
Compare the test statistic to the critical value. If the test statistic is less than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

In this case, the test statistic (-0.479) is greater than the critical value (-1.28). Therefore, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude, at the 0.1 level of significance, that the mean monthly rent for apartments on the east side is less than $675.

In summary, based on the given data and using a one-tailed test at the 0.1 level of significance, we cannot conclude that the mean monthly rent for apartments on the east side is less than $675.