Question

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 the p - 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	228	265	359	482	534
crash fatality rate	15.8	15.7	15.4	15.3	14.9

what are the null and alternative hypotheses?
○ a. $h_0: \
ho = 0$ $h_1: \
ho < 0$ ○ b. $h_0: \
ho \
eq 0$ $h_1: \
ho = 0$
○ c. $h_0: \
ho = 0$ $h_1: \
ho \
eq 0$ ○ d. $h_0: \
ho = 0$ $h_1: \
ho > 0$
construct a scatterplot. choose the correct graph below.
○ a. ○ b. ○ c. ○ d.
(graphs for scatterplot options are shown with y - axes from 15 to 17 and various data point distributions)

Explanation:

To determine the null and alternative hypotheses for testing linear correlation, we recall that the null hypothesis \( H_0 \) for a correlation test assumes no linear correlation (i.e., \(
ho = 0 \)), and the alternative hypothesis \( H_1 \) tests for a non - zero correlation (since we are checking if there is any linear correlation, positive or negative).

• Option A: \( H_1:

ho < 0 \) tests for a negative correlation only, but we want to test for any linear correlation (positive or negative), so A is incorrect.

• Option B: The null hypothesis \( H_0:

ho
eq0 \) is wrong because the null hypothesis should assume no correlation (\(
ho = 0 \)) initially, so B is incorrect.

• Option C: \( H_0:

ho = 0 \) (no linear correlation) and \( H_1:
ho
eq0 \) (there is a linear correlation, either positive or negative) is the correct set of hypotheses for testing if there is a linear correlation between two variables.

• Option D: \( H_1:

ho>0 \) tests for a positive correlation only, but we want to test for any linear correlation, so D is incorrect.

For the scatterplot, we analyze the relationship between lemon imports (x - values: 228, 265, 359, 482, 534, increasing) and crash fatality rate (y - values: 15.8, 15.7, 15.4, 15.3, 14.9, decreasing). So as x increases, y decreases, indicating a negative linear trend. Looking at the graphs, we need to find the one where as x (lemon imports) increases, y (crash fatality rate) decreases. From the visual description, graph B should show this negative trend (points going down as x increases).

Answer:

For the hypotheses: C. \( H_0:
ho = 0 \), \( H_1:
ho
eq0 \)
For the scatterplot: B. The Middle Graph (assuming B is the middle - looking graph with the negative trend)