The dataset HomesForSaleCA contains a random sample of 30...

The dataset HomesForSaleCA contains a random sample of 30 houses for sale in California. We are interested in whether there is a positive association between the number of bathrooms and number of bedrooms in each house. (a) What are the null and alternative hypotheses for testing the correlation? (b) Find the correlation in the sample. (c) Calculate (or use technology to find) the appropriate test statistic, and determine the p-value. (d) State the conclusion in context.

The dataset HomesForSaleCA contains a random sample of 30 houses for sale in California. We are interested in whether there is a positive association between the number of bathrooms and number of bedrooms in each house.
(a) What are the null and alternative hypotheses for testing the correlation?
(b) Find the correlation in the sample.
(c) Calculate (or use technology to find) the appropriate test statistic, and determine the p-value.
(d) State the conclusion in context.

Key Concepts

Pearson's Correlation Coefficient

This concept measures the strength and direction of the linear relationship between two continuous variables. It is a standardized measure that ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values near -1 indicate a strong negative linear relationship, and values around 0 suggest no linear association. In statistical analysis, calculating this coefficient is often the first step in exploring bivariate data relationships.

Null and Alternative Hypotheses

In hypothesis testing for correlation, the null hypothesis typically states that there is no linear relationship between the two variables in the population (i.e., the population correlation coefficient is zero). Conversely, the alternative hypothesis posits that there is a non-zero linear relationship, which in some cases can be directional (e.g., a positive association). These hypotheses provide the framework for determining whether the observed sample correlation is statistically significant.

Test Statistic for Correlation

The test statistic used to assess the significance of a sample correlation is derived from the Pearson’s correlation coefficient. Commonly, the statistic is calculated using a transformation that follows a t-distribution with n - 2 degrees of freedom, where n is the sample size. This statistic helps to determine whether the observed correlation could reasonably occur under the null hypothesis of no association.

P-value

The p-value in the context of correlation testing measures the probability of obtaining an observed correlation as extreme as, or more extreme than, the one calculated from the sample, under the assumption that the null hypothesis is true. A small p-value indicates strong evidence against the null hypothesis, suggesting that the correlation observed in the sample is unlikely to be due to random chance.

Statistical Inference in Context

Drawing conclusions in context involves interpreting the results of the hypothesis test in real-world terms. This means assessing whether the evidence is sufficient to support a claim about the existence and direction of the association between the variables. It is critical to consider the sample size, effect size, and the practical significance of the findings when making inferences that can inform decisions or further research.

Transcript

0:00 All right.

00:02 So this question deals with data collected about the number of bedrooms in renter -occupied homes versus owner -occupied homes.

00:11 So, part wants us to create a random variable x for the number of bedrooms in a renter -occupied home.

00:24 So, since they give us frequency instead of probability, we have to convey.

00:32 So remember that probability equals frequency over total.

00:44 So let's define our variable x right here, and it can be 0 through 4, and then we have p of x.

01:02 So to find our p of x, all we have to do is divide each entry in the table by the column total, which plugging that into a calculator, you'll get these numbers.

01:16 For 0 .0368, for 1, it's 0 .337.

01:25 For 2, it's 0 .412.

01:31 For 3, it's 0 .178.

01:35 And for 4, it's 0 .0375.

01:44 Okay.

01:47 So part b wants us to compute the expected value, and variance of x.

02:00 So, we know that the expected value of x equals the sum of x weighted by its probability.

02:13 So, looking at our chart, we know it's 0 times 0368 plus 1 times 0 .337 plus dot, dot, dot, dot, for all the values of x.

02:37 And that works out to 1 .84 bedrooms.

02:47 All right.

02:50 And then it also wants for the variance.

02:55 And that is the sum of each value's distance from the mean squared weighted by its probability and plugging in the numbers using our expected value from above you will find that the variance is equal to 0 .787.

03:19 7.

03:22 All right.

03:26 So part c now wants us to make a probability distribution for the owner occupied homes.

03:38 So this is the same thing as we did in part a just with different numbers...

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