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Chapter 13: Problem 36

When it comes to high-end Japanese eateries featuring sushi, the quality and presentation of the food are no doubt indicators of the cost. What about the d茅cor of the restaurant? Would that also have an effect on the cost? Zagat Survey results, published in Newsweek, produced a correlation coefficient of 0.532 between restaurant d茅cor rating and the average cost of dinner. If these results were based on five restaurants, can we conclude the relationship is significant at the 0.05 level of significance?

Short Answer

Expert verified
No, the relationship between restaurant d茅cor and the average cost of dinner is not statistically significant at the 0.05 level based on the provided information.

Step by step solution

01

Identify the values given and what needs to be found

We are given the correlation coefficient r = 0.532 and the number of observations n = 5. We need to find if the relationship defined by this correlation is statistically significant at the 0.05 significance level.
02

Calculate the critical value

The critical value for a two-tailed test at the 0.05 significance level and with degree of freedom (n-2) = 5 - 2 = 3, can be determined from a standard Critical Values of the Student's t Distribution table to be approximately 3.182. The Student's t-Distribution is used because we are dealing with a small sample size.
03

Calculate the t-value

The t-value is calculated using the correlation coefficient (r) and the number of observations (n) with the following formula: \( t = r/\sqrt{(1 - r^2)/(n-2)} \). Substituting the given values into this formula, we find that \( t = 0.532/\sqrt{(1 - 0.532^2)/(5-2)} = 0.779 \)
04

Compare the t-value to the critical value

Our calculated t-value (0.779) is less than our critical value (3.182), which means that we cannot reject the null hypothesis, i.e., we cannot consider the correlation between the restaurant's d茅cor rating and the average cost of dinner to be statistically significant at the 0.05 level of significance.

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Key Concepts

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

Understanding the Correlation Coefficient
The correlation coefficient is a crucial statistic for measuring the strength and direction of the linear relationship between two variables. In simpler terms, it tells us how likely it is that when one variable changes, the other one will too, and whether that change is in the same direction (positive correlation) or the opposite one (negative correlation).

Imagine it as a dance between two partners: if they move perfectly in synch, the correlation is strong; if they move randomly in respect to one another, the correlation is weak. The correlation coefficient

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Most popular questions from this chapter

Determine the \(p\) -value for each of the following situations. a. \(\quad H_{a}: \beta_{1}>0,\) with \(n=18\) and \(t \star=2.4\) b. \(\quad H_{a}: \beta_{1} \neq 0,\) with \(n=15, b_{1}=0.16,\) and \(s_{b_{1}}=0.08\) c. \(\quad H_{a}: \beta_{1}<0,\) with \(n=24, b_{1}=-1.29,\) and \(s_{b_{1}}=0.82\)

Consider a set of paired bivariate data. a. \(\quad\) Explain why \(\Sigma(x-\bar{x})=0\) and \(\Sigma(y-\bar{y})=0\). b. \(\quad\) Describe the effect that lines \(x=\bar{x}\) and \(y=\bar{y}\) have on the graph of these points. c. \(\quad\) Describe the relationship of the ordered pairs that will cause \(\Sigma[(x-\bar{x}) \cdot(y-\bar{y})]\) to be (1) positive, (2) negative, and (3) near zero.

Determine the bounds on the \(p\) -value that would be used in testing each of the following null hypotheses using the \(p\) -value approach: a. \(\quad H_{o}: \rho=0\) vs. \(H_{a}: \rho \neq 0,\) with \(n=32\) and \(r=0.41\) b. \(\quad H_{o}: \rho=0\) vs. \(H_{a}: \rho>0,\) with \(n=9\) and \(r=0.75\) c. \(\quad H_{o}: \rho=0\) vs. \(H_{a}: \rho<0,\) with \(n=15\) and \(r=-0.83\)

The test-retest method is one way of establishing the reliability of a test. The test is administered, and then, at a later date, the same test is readministered to the same individuals. The correlation coefficient is computed between the two sets of scores. The following test scores were obtained in a test-retest situation. $$\begin{array}{lllllllllll} \text { First Score } & 75 & 87 & 60 & 75 & 98 & 80 & 68 & 84 & 47 & 72 \\ \hline \text { Second Score } & 72 & 90 & 52 & 75 & 94 & 78 & 72 & 80 & 53 & 70 \end{array}$$ Find \(r\) and set a \(95 \%\) confidence interval for \(\rho\).

State the null hypothesis, \(H_{o},\) and the alternative hypothesis, \(H_{a},\) that would be used to test the following statements: a. The slope for the line of best fit is positive. b. The slope of the regression line is not significant. c. The negative slope for the regression is significant.
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