A random sample of n 1 = 10 regions in New England gave the following violent cr

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

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

Assume that the crime rate distribution is approximately normal in both regions. Note: If a two-sample t-test is appropriate, for degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

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 ? = 0.01.(i) What is the level of significance?


State the null and alternate hypotheses.

H0: ?1 = ?2; H1: ?1 ? ?2H0: ?1 = ?2; H1: ?1 < ?2    H0: ?1 ? ?2; H1: ?1 = ?2H0: ?1 < ?2; H1: ?1 = ?2H0: ?1 = ?2; H1: ?1 > ?2

(ii) 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 Student's t. 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 standard normal. 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 ?1 ? ?2. Round your answer to three decimal places.)


(iii) Find the P-value. (Round your answer to four decimal places.)


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

(iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ??

Since the P-value ? ?, we fail to reject H0. The data are statistically significant.Since the P-value > ?, we fail to reject H0. The data are not statistically significant.    Since the P-value > ?, we reject H0. The data are not statistically significant.Since the P-value ? ?, we reject H0. The data are statistically significant.

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

Reject H0. At the 1% level of significance, the evidence is sufficient to indicate that violent crime in the Rocky Mountain region is higher than in New England.Reject H0. At the 1% level of significance, the evidence is insufficient to indicate that violent crime in the Rocky Mountain region is higher than in New England.    Fail to reject H0. At the 1% level of significance, the evidence is insufficient to indicate that violent crime in the Rocky Mountain region is higher than in New England.Fail to reject H0. At the 1% level of significance, the evidence is sufficient to indicate that violent crime in the Rocky Mountain region is higher than in New England.

(b) Find a 98% confidence interval for ?1 ? ?2. (Round your answers to two decimal places.)

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

At the 98% level of confidence, we can conclude that the population mean violence rates are different.At the 98% level of confidence, we cannot conclude that the population mean violence rates are different.   

Explanation / Answer

x1 = 3.5556

x2 = 3.8833

s1 = 0.9207

s2 = 0.9675

a)

H0: mu1 = mu2

Ha: mu1 < mu2

ii)

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

SE = sqrt[ (s12/n1) + (s22/n2) ]

(s12/n1) = 0.0942

(s22/n2) = 0.0780

SE = 0.4150

Test statistic,

t = [ (x1 - x2) - d ] / SE

t = -0.7899

iii)

p-value = 0.2250

sketch - Option 1

iv)

Since the P-value > ?, we fail to reject H0. The data are not statistically significant.

v)

Fail to reject H0. At the 1% level of significance, the evidence is insufficient to indicate that violent crime in the Rocky Mountain region is higher than in New England.

b)

x1bar - x2bar = -0.33

SE = 0.4150

CI = 98%

DF = 9

t-value = 2.8214

ME = t*SE = 1.1708

Confidence Interval, (x1bar - x2bar) +/- ME

Lower bound = -1.49860

Upper bound = 0.84304

Confidence Interval = (-1.4986 , 0.843 )

At the 98% level of confidence, we cannot conclude that the population mean violence rates are different.

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