Hi, Julia! I hope this explanation helps you understand and answer your question.
IMPORTANT NOTE: For this problem, you would need to conduct a two-sided z-test, as opposed to a two-sided t-test. The main indicator for that is that the problem gives us the population standard deviation. The procedure you would typically follow may differ from mine, but I like to use the acronym, "PHANTOMS" when I conduct hypothesis tests. Define your parameters. State your hypotheses. Assess the conditions. Name the test. Calculate the test statistic. Obtain a P-value. Make a decision. State your conclusion in context.
a.) Your parameter would be defined by μ, which represents the true mean home price, in thousands of dollars, for this year. Your null hypothesis would be μ = 260.7, while your alternative hypothesis would be μ≠ 260.7.
b.) For this problem, the appropriate equation to use to calculate the z-statistic would be z = (x̅ - μ) / (σ/√n). Your sample mean x̅ is 290.5, your population mean is 260.7, your standard deviation σ is 150, and your sample size n is 50. Plug those numbers into the equation to get the z-statistic. You should get an answer of approximately 1.4048.
c.) Using normalcdf (lower, upper, μ, σ ) on your graphing calculator, enter the following numbers in this order: normalcdf (1.4048, 1E99, 0, 1). Your P-value should come out to approximately 0.08. BUT YOU'RE NOT DONE YET! Remember, this is a two-sided z-test, so you need to multiply your P-value by 2 to get a final answer of 0.16.
d.) The problem indicates that the significance level we are supposed to compare our P-value to is 0.05. Therefore, since our P-value is greater than alpha (0.16 > 0.05), we fail to reject the null hypothesis.
e.) Now, the final part: state your conclusion in context. At 5% significance, we do not have enough evidence to conclude that the mean home price, in thousands of dollars, for this year differs from the mean price in April through June of 2012.