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Penfield and Perinton are two adjacent eastside suburbs of Rochester, New York. They have both always been considered on equal ground with respect to quality of life, housing, and education. Although many new housing developments are going up in Penfield, Perinton offers the added bonus of cheaper energy rates. It is thought that Perinton is able to have higher asking prices because of this bonus. To test this theory, random samples of real estate transactions were taken in each suburb during the week of October 17 \(2009 .\) Do the data support the theory for this time frame? Use \(\alpha=0.10\) $$\begin{array}{lc} \text { Pentield } & \text { Perinton } \\ \hline 195.700 & 154,900 \\\137,500 & 429,000 \\\117,000 & 272,000 \\\115,000 & 160,609 \\\176,000 & 265,000 \\\149,013 & 144,000 \\\130,000 & 152,000 \\ 266,490 & 390,000 \\\152,000 & 130,300 \\\262,765 & 149,900 \\\\\hline\end{array}$$

Step by step solution

01

Calculate Sample Means

Given the data, find the average (mean) transaction price for each suburb (Penfield and Perinton). This can be calculated using the formula for the mean, which is the sum of the values divided by the number of values.

02

Conduct Hypothesis Test

Formulate the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis is that the means of the two suburbs are equal, while the alternative hypothesis is that the mean price of Perinton is greater than that of Penfield. Then, perform a two-sample t-test at significance level \(\alpha = 0.10\)

03

Calculate the T-Statistics and P-value

After the hypotheses have been stated, compute the T-Statistic and the P-value using a statistical software or calculator. These values will analyse the significant difference between the means of the two groups.

04

Make the Decision and Interpret Results

Check if the P-value is less than the significance level (0.10). If it is, reject the null hypothesis and conclude that there is enough evidence to support the theory that the mean prices in Perinton are higher. If the P-value is greater than the significance level, fail to reject the null hypothesis and conclude that there is not sufficient evidence to support this theory.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Sample t-Test

A two-sample t-test is an essential statistical method used when comparing the means of two groups to determine if they are significantly different from each other. In our scenario, it helps to compare the average real estate prices in the two suburbs. When researchers suspect that there might be a difference between two independent groups, this test becomes very helpful. It works by analyzing the data and testing the means of the two samples.

• To perform a two-sample t-test, we first calculate the sample means and standard deviations for each group.
• We use these values to calculate the t-statistic which tells us how far apart the two sample means are, relative to the variability in the samples.
• This t-statistic is then compared to a critical value from the t-distribution table that corresponds to the chosen significance level.

This method places emphasis on assessing whether any observed difference is likely due to real effects rather than random chance.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference, serving as a baseline to compare against. In the context of the two-sample t-test for our exercise, the null hypothesis states that there is no difference in the mean housing prices between Penfield and Perinton.

• The purpose of the null hypothesis is to assume no effect or no difference until proven otherwise.
• It's a starting point that allows the assessment of whether observed data could happen by chance alone.
• Rejecting the null hypothesis means you have enough evidence to say the groups are different, while failing to reject it suggests no strong evidence against \(H_0\).

Without a null hypothesis, it would be difficult to provide a meaningful conclusion from the test results.

Alternative Hypothesis

The alternative hypothesis, symbolized as \(H_1\) or sometimes \(H_a\), offers a counterpoint to the null hypothesis. It is what researchers aim to prove in their study. In the context of the discussed exercise, the alternative hypothesis might state that the mean real estate prices in Perinton are higher than those in Penfield.

• The formulation of the alternative hypothesis is crucial because it defines the direction of the test. It can be two-sided or one-sided, depending on what you are testing for.
• Here, it's a one-sided test, since we are only interested in seeing if Perinton's prices are greater, not merely different.
• Proving the alternative hypothesis involves collecting evidence that convincingly contradicts the null hypothesis.

Properly defining the alternative hypothesis helps in determining the direction and nature of your hypothesis testing.

Significance Level

The significance level, denoted as \(\alpha\), is a critical threshold in hypothesis testing that helps decide if we can reject the null hypothesis. It's typically set before conducting the test, and it represents the probability of making a Type I error, which is rejecting the null hypothesis when it's actually true. In our case, a significance level of 0.10 was chosen.

• Choosing \(\alpha = 0.10\) indicates a 10% risk of concluding there is a significant difference when there is none.
• This level impacts the critical value from the t-distribution that your t-statistic is compared against.
• A lower \(\alpha\) means stricter criteria for rejecting \(H_0\), leading to fewer false positives, but may also reduce sensitivity to detect real effects.

The significance level is a key parameter in deciding the threshold for making decisions based on your test results.