Refer to the Real Estate data, which reports information on...

Refer to the Real Estate data, which reports information on the homes sold in Denver, Colorado, last year. a. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes with a pool and homes without a pool? b. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes with an attached garage and homes without a garage? c. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes in Township 1 and Township $2 ?$ d. Find the median selling price of the homes. Divide the homes into two groups, those that sold for more than (or equal to) the median price and those that sold for less. Is there a difference in the proportion of homes with a pool for those that sold at or above the median price versus those that sold for less than the median price? Use the .05 significance level.

Refer to the Real Estate data, which reports information on the homes sold in Denver, Colorado, last year.
a. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes with a pool and homes without a pool?
b. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes with an attached garage and homes without a garage?
c. At the .05 significance level, can we conclude that there is a difference in the mean selling price of homes in Township 1 and Township $2 ?$
d. Find the median selling price of the homes. Divide the homes into two groups, those that sold for more than (or equal to) the median price and those that sold for less. Is there a difference in the proportion of homes with a pool for those that sold at or above the median price versus those that sold for less than the median price? Use the .05 significance level.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical process used to make inferences about a population parameter based on sample data. It involves formulating a null hypothesis, which represents no effect or no difference, and an alternative hypothesis, which reflects the effect or difference being tested. By analyzing the data via an appropriate test statistic and comparing it to a critical value or assessing the p-value, researchers determine whether to reject the null hypothesis or not.

Two-Sample t-Test

The two-sample t-test is used to compare the means of two independent groups to determine if there is a statistically significant difference between them. This test assumes that the populations from which the samples are drawn are normally distributed and, depending on the variant, may assume equal variances. It is particularly useful when sample sizes are small and when comparing the means of two different conditions or groups.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used in hypothesis testing to decide whether to reject the null hypothesis. A significance level of 0.05, for example, implies that there is a 5% risk of rejecting the null hypothesis when it is actually true. It helps in controlling the probability of committing a Type I error in statistical tests.

Central Tendency (Median)

The median is a measure of central tendency that represents the middle value of a data set when the values are arranged in order. It divides the data into two equal halves, with 50% of the observations falling below the median and 50% above it. It is especially useful in distributions that are skewed or contain outliers, as it is less affected by extreme values compared to the mean.

Test for Difference in Proportions

A test for difference in proportions is used to compare the proportion of a particular outcome between two distinct groups. This typically involves using the two-proportion z-test to determine if the observed difference in proportions is statistically significant, which is important when assessing whether a categorical variable behaves differently across groups defined by another criterion.

Transcript

00:01 In this question, we want to compare house prices in the south and in the midwest.

00:06 So we sampled some house in both region and we want to know if the price difference is greater than $3 ,400.

00:17 So let's get started.

00:19 We're going to use hypothesis testing in the z test, more specifically presented at page 491 of your book.

00:26 Step 1, we gather all of our data saw both, not in, both in the south and in the midwest, 40 houses were sample.

00:40 In the south, the average house price in the samples was $261 ,500, whereas in the midwest it was $261 ,500, whereas in the midwest it was $200 ,000.

00:56 $148 ,200.

01:02 The standard deviation for the population, so in the south is $110 ,500, no, not $110, sorry, $10 ,000, $10 ,500.

01:19 So let me just erase an extra one.

01:23 And it's the standard deviation in the midwest is $248 ,000.

01:30 No, i'm writing the mean of the sample.

01:37 Silly me.

01:39 Okay, so the standard deviation for houses in the whole midwest is $12 ,000.

01:47 Great.

01:49 We wrote down our data.

01:51 We made tiny mistakes along the way.

01:53 We fixed them.

01:54 Perfect.

01:56 Now we read to write down some hypotheses.

01:59 So our first hypothesis will be that the difference in price in the south and in the in the midwest is equal to $3 ,400 or $3 ,400 if you prefer to say it like that.

02:17 Our alternative hypothesis will be the difference in price in the south and in the midwest is greater than $3 ,400.

02:27 So this is the claim of the question, which basically means we're going to have a two -tailed test.

02:37 So let's say this is a normal...

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