7.41 Murders and poverty, Part II. Exercise 7.29 presents regression output from a model for predicting annual murders per million from percentage living in poverty based on a random sample of 20 metropolitan areas. The model output is also provided below.

Estimate Std. Error t value Pr(>|t|)
(Intercept) -29.901 7.789 -3.839 0.001
poverty% 2.559 0.390 6.562 0.000

s = 5.512      R2 = 70.52%     R2adj = 68.89%$

  1. What are the hypotheses for evaluating whether poverty percentage is a significant predictor of murder rate?
    H0 = The poverty percentage is NOT a significant predictor of murder rate.
    HA = The poverty percentage is a significant predictor of murder rate.

    More formally the hypothesis could be expressed
    H0 : \(\beta\)1 = 0
    HA : \(\beta\)1 \(\neq\) 0

  2. State the conclusion of the hypothesis test from part (a) in context of the data.

    Based on the summary chart above, I would say that the null hypothesis should be rejected as poverty is a significant predictor of murder rate. I base this on the Pr value of the Poverty line which has a probabilty of essentially zero that the poverty predictor is purely chance.

  3. Calculate a 95% confidence interval for the slope of poverty percentage, and interpret it in context of the data.

    The 95% confidence interval is calculated using a T Distribution, with degrees of freedom of 18 (as there are 20 metropolitan areas, less 2 sided test), and the equation (using the t-value from the t probability table) is essentially: poverty pct \(\pm\) (2.10 * SE), giving (1.74, 3.38). The more exact “R” calculation is below. This interval basically means that each increase of one percent in poverty , will, within 95 confidence, increase the number of murders per million in the range of 1.74 to 3.38.

2.559 + (qt(.975, df = 18) * .39)
## [1] 3.37836
2.559 - (qt(.975, df = 18) * .39)
## [1] 1.73964

  1. Do your results from the hypothesis test and the confidence interval agree? Explain.

    Yes they do agree. The result from the hypothesis test was to reject the null hypothesis and say that poverty is a predictor. Since the confidence interval does not cross the zero boundary, it also implies that povery is a predictor of the murder rate.