Mathematics, 05.12.2019 07:31 granthazenp5e9mj

Listed below are annual data for various years. the data are weights (metric tons) of imported lemons and car crash fatality rates per 100,000 population. construct a scatterplot, find the value of the linear correlation coefficient r, and find thep-value using α=0.05. is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality rates? do the results suggest that imported lemons cause car fatalities?

lemon_imports_(x) crash_fatality_rate_(y)

230 15.8

264 15.6

359 15.5

482 15.3

531 14.9

what are the null and alternative hypotheses?

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Answer from: kiera2599

Step-by-step explanation:

In statistics, correlations are used to find out how strong is the relation between two variables. This is applied in experimental environments to prove a hypothesis through this relation. So, this correlation is called Pearson's R or Pearson's correlation.

The Pearson's R is calculated with the formula attached, where x and y are the values given. So, applying the equation, the correlation is -0.94, which is a negative correlation, in other words, as lemon imports rises, crash fatality rates decreases, that's the relation between these two variables.

To know how to calculate the R correlation, first we need to build the table for the X and Y values attached. Then. we applied the formula:

$r=\frac{5(28603.4)-(1866)(77.1)}{\sqrt{(5(765762)-1866^{2})(5(1189.35)-77.1^{2} ) } } \\r=\frac{143017-143868.6}{\sqrt{(346854)(2.34)} } \\r=\frac{-851.6}{900.9}= -0.94$

Then, we calculate the t value:

$t=\frac{r\sqrt{n-2} }{\sqrt{1-r^{2} } } = \frac{-0.94\sqrt{5-2} }{1-(-0.94)^{2} } \\t=\frac{-0.94(1.73)}{0.34} =4.78$

So, using a degree of freedom of 9 (df=9), the probability (P-value) is less than 0.05 which is the level of significance.

Therefore, if the P-value is smaller than the level of significance, the null hypothesis must be reject, that is, the correlation exists. In other words, there's a linear correlation between lemon imports and crash fatality rates. However, this correlation doesn't implies a causation, which means that not always the lemon imports increases, the fatality rates decreases. And, if there's no causation determined, we cannot generalize the results.

Therefore, the results don't exactly say that imported lemons cause car fatalities. Actually, points to the opposite direction. It's probable that lemons importation don't cause car fatalities.

In this problem, the null hypothesis is "t > 0.05", and the alternative hypothesis is "t < 0.05".

Specifically, this null hypothesis says "there's no enough evidence to find a correlation". The alternative hypothesis says "there's enough evidence to have a correlation".

The scatterplot is attached, there you will see that all data is closely related and tend to an linear behaviours, from there you already can imagine that the correlation exists.