Question

A random sample of n1 = 10 regions in New England gave the following violent crime rates (per million population). x1: New England Crime Rate 3.5 3.7 4.2 3.9 3.3 4.1 1.8 4.8 2.9 3.1 Another random sample of n2 = 12 regions in the Rocky Mountain states gave the following violent crime rates (per million population). x2: Rocky Mountain Crime Rate 3.5 4.3 4.5 5.1 3.3 4.8 3.5 2.4 3.1 3.5 5.2 2.8 Assume that the crime rate distribution is approximately normal in both regions. (i) Use a calculator to calculate x1, s1, x2, and s2. (Round your answers to two decimal places.) x1 = s1 = x2 = s2 = (ii) Do the data indicate that the violent crime rate in the Rocky Mountain region is higher than in New England? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ1 = μ2; H1: μ1 < μ2 H0: μ1 = μ2; H1: μ1 ≠ μ2 H0: μ1 < μ2; H1: μ1 = μ2 H0: μ1 = μ2; H1: μ1 > μ2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume that both population distributions are approximately normal with known standard deviations. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. What is the value of the sample test statistic? (Test the difference μ1 − μ2. Round your answer to three decimal places.) (c) Find (or estimate) the P-value. P-value > 0.250 0.125 < P-value < 0.250 0.050 < P-value < 0.125 0.025 < P-value < 0.050 0.005 < P-value < 0.025 P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value. Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. (e) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is sufficient evidence that violent crime in the Rocky Mountain region is higher than in New England. Fail to reject the null hypothesis, there is insufficient evidence that violent crime in the Rocky Mountain region is higher than in New England. Reject the null hypothesis, there is insufficient evidence that violent crime in the Rocky Mountain region is higher than in New England. Fail to reject the null hypothesis, there is sufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.

Answer 1

I used R software to solve this question.

R codes and output:

x1=c(3.5,3.7,4.2,3.9,3.3,4.1,1.8,4.8,2.9,3.1)
> x2=c(3.5,4.3,4.5,5.1,3.3,4.8,3.5,2.4,3.1,3.5,5.2,2.8)
> mean(x1)
[1] 3.53
> sd(x1)
[1] 0.8287206
> mean(x2)
[1] 3.833333
> sd(x2)
[1] 0.9217901

Que. i

and s₁ = 0.83

and s₂ = 0.92

Que.ii

a)

Level of significance is the probability of rejecting null hypothesis when it is true. Here it is 0.01

Hypothesis:

H₀ : ; H₁ :

b)

The student's t. We assume that both population distribution are approximately normal with unknown standard deviation.

c)

t.test(x1,x2, alternative="less")

Welch Two Sample t-test

data: x1 and x2
t = -0.81219, df = 19.856, p-value = 0.2132
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
-Inf 0.3410378
sample estimates:
mean of x mean of y
3.530000 3.833333

Test statistics value = -0.812

0.125 < p - value < 0.250

d)

At the level, we fail t reject null hypothesis and conclude that data are not statistically significant.

e)

Fail to reject null hypothesis, there is insufficient evidence that violent crime the Rocky mountain region is higher than in New England.