Pat wants to compare the cost of one- and two-bedroom apartments in the area of her college campus. She collects data for a random sample of 10 advertisements of each type. The table below shows the rents (in dollars per month) for the selected apartments.

Pat wonders if two-bedroom apartments rent for significantly more, on average than one-bedroom apartments. She decides to perform a test of $H_0:{\mu }_1={\mu }_2$ versus $H_a:{\mu }_1<{\mu }_2$, where ${\mu }_1$ and ${\mu }_2$ are the true mean rents for all one-bedroom and two-bedroom aparaments, respectively, near the campus.

(a) Name the appropriate test and show that the conditions for carrying out this test are met.

(b) The appropriate test from part (a) yields a P-value of $0.058$. Interpret this P-value in context.

(c) What conclusion should Pat draw at the$\alpha =0.05$ significance level? Explain.

In general:

One proportion: one-sample $z$test/interval

Two proportions: two-sample $z$test/interval

One means: one-sample $t$test/interval

Two means: two-sample $t$test/interval or paired $t$test/interval

Use a test if you want to test for a difference, equality, increase or decrease. Use an interval if you want to estimate an interval in which the true value lies.

The two-sample t test for a population mean the difference

because we are interested in testing the means of two samples.

There are three conditions for the test: Random, Normal, Independent

Random: Satisfied, because the samples are given to be random samples.

Normal: Satisfied, because the histograms of the samples do not show strong skewness or outliers.

Independent: Satisfied, because the sample size is less than $10\%$of the population size.

Thus we note that all three conditions have been satisfied.

$H_0:{\mu }_1={\mu }_2$

$H_a:{\mu }_1<{\mu }_2$

$P=0.058=5.8\%$

The $P$-value is the probability of obtaining the value of the test statistic, or a value more extreme if the null hypothesis $H_0$is true.

This then means that if the population means are equal, then the probability of obtaining two samples with a difference as in the given two samples (or a more extreme difference) is localid="1650520353140" $5.8\%.$

$H_0:{\mu }_1={\mu }_2$

$H_a:{\mu }_1<{\mu }_2$

$P=0.058=5.8\%(\text{Exercise}13\mathrm{b})$

If the P-value is smaller than the significance level, reject the null hypothesis:

$P=0.058>0.05\Rightarrow \text{Fail to reject}{H}_0$

There is not sufficient evidence to support the claim that two-bedroom apartments rent for significantly more, on average than one-bedroom apartments.