Construct a scatterplot, and find the value of the linear...

Construct a scatterplot, and find the value of the linear correlation coefficient $r$ Also find the $P$ -value or the critical values of $r$ from Table $A$ - $6 .$ Use a significance level of $\alpha=0.05 .$ Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section $10-2$ exercises.)The "pizza connection" is the principle that the price of a slice of pizza in New York City is always about the same as the subway fare. Use the data listed below to determine whether there is a significant linear correlation between the cost of a slice of pizza and the subway fare.$$\begin{array}{l|l|l|l|l|l|l|l|l|l}\hline \text { Year } & 1960 & 1973 & 1986 & 1995 & 2002 & 2003 & 2009 & 2013 & 2015 \\\hline \text { Pizza cost } & 0.15 & 0.35 & 1.00 & 1.25 & 1.75 & 2.00 & 2.25 & 2.30 & 2.75 \\\hline \text { Subway Fare } & 0.15 & 0.35 & 1.00 & 1.35 & 1.50 & 2.00 & 2.25 & 2.50 & 2.75 \\\hline \text { CP1 } & 30.2 & 48.3 & 112.3 & 162.2 & 191.9 & 197.8 & 214.5 & 233.0 & 237.2 \\\hline\end{array}$$

Key Concepts

Scatterplot

A scatterplot is a graphical representation of data using Cartesian coordinates, where each point represents an observation's values for two variables. It is used to visually inspect the relationship between variables, identify patterns, trends, and potential outliers, and to assess whether a linear model might be appropriate.

Linear Correlation Coefficient (r)

The linear correlation coefficient, denoted by r, is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables. Its value ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values near 0 indicate little to no linear association.

Hypothesis Testing for Correlation

This concept involves setting up a null hypothesis that there is no linear relationship (r = 0) between the two variables, and an alternative hypothesis that a linear correlation exists. The test assesses whether the observed correlation is statistically significant, meaning it is unlikely to have occurred by chance.

P-value and Critical Values

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis is true. Critical values, obtained from statistical tables, represent the threshold beyond which the null hypothesis is rejected. Both are used to determine whether the evidence is strong enough to conclude that a significant linear relationship exists.

Significance Level (?)

The significance level, denoted as ?, is a threshold used in hypothesis testing to measure the risk of rejecting a true null hypothesis. Commonly set at 0.05, it defines the probability of a Type I error, where the investigator concludes that there is a significant effect or relationship when in fact there is none.

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