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

4.0

4.1

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

3.7

4.3

4.7

5.3

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. Use a calculator to calculate x₁, s₁, x₂, and s₂. (Round your answers to two decimal places.)

x1	=
s1	=
x2	=
s2	=

(a) Do the data indicate that the violent crime rate in the Rocky Mountain region is higher than in New England? Use α = 0.01.(i) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂H₀: μ₁ = μ₂; H₁: μ₁ > μ₂    H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂H₀: μ₁ < μ₂; H₁: μ₁ = μ₂

(ii) What sampling distribution will you use? What assumptions are you making?

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 known standard deviations.    The standard normal. We assume that both population distributions are approximately normal with unknown 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? Compute the corresponding z or t value as appropriate. (Test the difference μ₁ − μ₂. Do not use rounded values. Round your answer to three decimal places.)


(iii) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(iv) Based on your answers in parts (i)−(iii), 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 statistically significant.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 statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(v) Interpret your conclusion in the context of the application.

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.    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 sufficient evidence that violent crime in the Rocky Mountain region is higher than in New England.

(b) Find a 98% confidence interval for

μ₁ − μ₂.

(Round your answers to two decimal places.)

lower limit
upper limit

Explain the meaning of the confidence interval in the context of the problem.

Because the interval contains only positive numbers, this indicates that at the 98% confidence level, the population mean violence rate in the Rocky Mountain region is lower than in New England.Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we cannot say that the population mean violence rate in the Rocky Mountain region is higher than in New England.    Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, the population mean violence rate in the Rocky Mountain region is lower than in New England.Because the interval contains only negative numbers, this indicates that at the 98% confidence level, the population mean violence rate in the Rocky Mountain region is higher than in New England.

Answer 1

a)

from above:

x1 =3.53

s1 =0.83

x2 =3.88

s2=0.96

level of significance =0.01

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

std error =√(S21/n1+S22/n2)=		0.3802
test stat t =(x1-x2-Δo)/Se =		-0.929

iii)

0.125 < P-value < 0.250

iv)

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

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

b)

Point estimate of differnce =x1-x2     =			-0.353
for 98 % CI & 9 df value of t=			2.821
margin of error E=t*std error                   =			1.073
lower bound=mean difference-E     =			-1.43
Upper bound=mean differnce +E      =			0.72

Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we cannot say that the population mean violence rate in the Rocky Mountain region is higher than in New England.