Question

In: Statistics and Probability

The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan...

The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan areas. The data are shown in the following table. Do these data provide evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = .05? (Give your answers correct to three decimal places.)

Population 10.3 0.6 4 6.3 4.4 6.5 5.4 1.5 4.6 3.9
Crime Rate 11.1 7.1 7 7 7.4 8.7 8.7 8.1 8.8 9.5

(a) Calculate r.


(ii) Calculate the critical region.
____ (smaller value)
____(larger value)

Solutions

Expert Solution

a)

The provided data are shown in the table below

X Y
10.3 11.1
0.6 7.1
4 7
6.3 7
4.4 7.4
6.5 8.7
5.4 8.7
1.5 8.1
4.6 8.8
3.9 9.5

Also, the following calculations are needed to compute the correlation coefficient:

X Y X*Y X2 Y2
10.3 11.1 114.33 106.09 123.21
0.6 7.1 4.26 0.36 50.41
4 7 28 16 49
6.3 7 44.1 39.69 49
4.4 7.4 32.56 19.36 54.76
6.5 8.7 56.55 42.25 75.69
5.4 8.7 46.98 29.16 75.69
1.5 8.1 12.15 2.25 65.61
4.6 8.8 40.48 21.16 77.44
3.9 9.5 37.05 15.21 90.25
Sum = 47.5 83.4 416.46 291.53 711.06

The correlation coefficient rr is computed using the following expression:

where

In this case, based on the data provided, we get that

Therefore, based on this information, the sample correlation coefficient is computed as follows

which completes the calculation.

The following needs to be tested:

H0​:ρ=0

HA​:ρ̸​=0

where \rhoρ corresponds to the population correlation.

The sample size is n = 10 , so then the number of degrees of freedom is df = n-2 = 10 - 2 = 8

The corresponding critical correlation value r_c for a significance level of α=0.05, for a two-tailed test is:

r_c = 0.632

Observe that in this case, the null hypothesis H0​:ρ=0 is rejected if |r| > r_c = 0.632

Based on the sample correlation provided, we have that |r| = 0.635 > r_c = 0.632, from which is concluded that the null hypothesis is rejected.

(ii) Calculate the critical region.
-0.632 (smaller value)
0.632 (larger value)


Related Solutions

The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan...
The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan areas. The data are shown in the following table. Do these data provide evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = .05? (Give your answers correct to three decimal places.) Population 9.9 2 1 7.2 0.6 6.6 0.6 7 6.5 0.6 Crime Rate 11.3 9 7.5 8.8 7.3 7.2 7.4 6.7 7.7 6.8...
The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan...
The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan areas. The data are shown in the following table. Do these data provide evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = .05? (Give your answers correct to three decimal places.) Population 10.9 2.6 4.3 7.6 3.5 3 3.3 6.4 0.7 3.4 Crime Rate 12.7 6.5 8.7 7.7 7.4 7.4 6.6 6.8 9.1 8.9...
The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan...
The population (in millions) and the violent crime rate (per 1000) were recorded for 10 metropolitan areas. The data are shown in the following table. Do these data provide evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = .05? (Give your answers correct to three decimal places.) Population 9.7 0.2 3.2 7.2 0.5 3.7 4.8 3.3 2.2 3.5 Crime Rate 13 8.7 8.8 9.2 6.8 8.4 9.3 7 7.1 6.6...
1) Using the excel data file “US violent crime” which shows the violent crime rate in...
1) Using the excel data file “US violent crime” which shows the violent crime rate in the US from 1960 to 2012: (20 pts) Make a time series plot of the data (5 pts each 25 pts total) Determine the following: Mean, Median, Standard deviation, Q1 and Q3. (25 pts) Make a histogram of the data. Hint the year is not used, you need to determine how many years fall into each of the classes. (7) What are your thoughts...
Does prison really deter violent crime? Let x represent percent change in the rate of violent...
Does prison really deter violent crime? Let x represent percent change in the rate of violent crime and y represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained. x 5.9 5.7 4.1 5.2 6.2 6.5 11.1 y −1.8 −4.1 −7.0 −4.0 3.6 −0.1 −4.4 A.given Σx = 44.7, Σy = −17.8,Σx2 = 315.05, Σy2 = 117.38, Σxy = −110.66, and r ≈ 0.065. Verify the given...
The following data represent crime rates per 1000 population for a random sample of 46 Denver...
The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.† 63.2 36.3 26.2 53.2 65.3 32.0 65.0 66.3 68.9 35.2 25.1 32.5 54.0 42.4 77.5 123.2 66.3 92.7 56.9 77.1 27.5 69.2 73.8 71.5 58.5 67.2 78.6 33.2 74.9 45.1 132.1 104.7 63.2 59.6 75.7 39.2 69.9 87.5 56.0 154.2 85.5 77.5 84.7 24.2 37.5 41.1 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x...
The following data represent crime rates per 1000 population for a random sample of 46 Denver...
The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.† 63.2 36.3 26.2 53.2 65.3 32.0 65.0 66.3 68.9 35.2 25.1 32.5 54.0 42.4 77.5 123.2 66.3 92.7 56.9 77.1 27.5 69.2 73.8 71.5 58.5 67.2 78.6 33.2 74.9 45.1 132.1 104.7 63.2 59.6 75.7 39.2 69.9 87.5 56.0 154.2 85.5 77.5 84.7 24.2 37.5 41.1 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x...
The following data represent crime rates per 1000 population for a random sample of 46 Denver...
The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.† 63.2 36.3 26.2 53.2 65.3 32.0 65.0 66.3 68.9 35.2 25.1 32.5 54.0 42.4 77.5 123.2 66.3 92.7 56.9 77.1 27.5 69.2 73.8 71.5 58.5 67.2 78.6 33.2 74.9 45.1 132.1 104.7 63.2 59.6 75.7 39.2 69.9 87.5 56.0 154.2 85.5 77.5 84.7 24.2 37.5 41.1 (1) Use a calculator with mean and sample standard deviation keys to find the sample mean x...
The following data represent crime rates per 1000 population for a random sample of 46 Denver...
The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.† 63.2 36.3 26.2 53.2 65.3 32.0 65.0 66.3 68.9 35.2 25.1 32.5 54.0 42.4 77.5 123.2 66.3 92.7 56.9 77.1 27.5 69.2 73.8 71.5 58.5 67.2 78.6 33.2 74.9 45.1 132.1 104.7 63.2 59.6 75.7 39.2 69.9 87.5 56.0 154.2 85.5 77.5 84.7 24.2 37.5 41.1 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x...
The following data represent crime rates per 1000 population for a random sample of 46 Denver...
The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.† 63.2 36.3 26.2 53.2 65.3 32.0 65.0 66.3 68.9 35.2 25.1 32.5 54.0 42.4 77.5 123.2 66.3 92.7 56.9 77.1 27.5 69.2 73.8 71.5 58.5 67.2 78.6 33.2 74.9 45.1 132.1 104.7 63.2 59.6 75.7 39.2 69.9 87.5 56.0 154.2 85.5 77.5 84.7 24.2 37.5 41.1 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT