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 ƒÏ = 0 in favor of ƒÏ �‚ 0 at ƒ¿ = .05? (Give your answers correct to three decimal places.)
Population 9.2 4.3 3.4 8 5.3 5.1 0.3 3.3 5.5 3.8
Crime Rate 12.9 7.6 9 7.4 9.2 6.6 7.7 8.1 6.5 8.4
(a) Calculate r.
Correct: 0.444 .
(ii) Calculate the critical region.
(smaller value)??
(larger value)??
F test
are you test F statistic?
I tried the F test calculator and that does not give me the answers that are needed. I have looked for one for critical region and have been unable to find one at this time.
F statistic = 1.866
F cirtical = 5.32
Sorry this is not right
Are you two tail
I think so
F statistic =1.854
Cirtical F value = 4.026
To calculate the Pearson correlation coefficient (r), you need to use the following formula:
r = (Σxy - (Σx * Σy) / n) / (√((Σx^2 - (Σx)^2 / n) * (Σy^2 - (Σy)^2 / n)))
where
Σxy is the sum of the product of the x and y values,
Σx is the sum of all x values,
Σy is the sum of all y values,
n is the number of data points.
Using the given data:
Population: 9.2, 4.3, 3.4, 8, 5.3, 5.1, 0.3, 3.3, 5.5, 3.8
Crime Rate: 12.9, 7.6, 9, 7.4, 9.2, 6.6, 7.7, 8.1, 6.5, 8.4
Let's calculate each part of the formula step by step:
Σxy = (9.2 * 12.9) + (4.3 * 7.6) + (3.4 * 9) + (8 * 7.4) + (5.3 * 9.2) + (5.1 * 6.6) + (0.3 * 7.7) + (3.3 * 8.1) + (5.5 * 6.5) + (3.8 * 8.4)
= 118.08 + 32.68 + 30.6 + 59.2 + 48.76 + 33.66 + 2.31 + 26.73 + 35.75 + 31.92
= 379.19
Σx = 9.2 + 4.3 + 3.4 + 8 + 5.3 + 5.1 + 0.3 + 3.3 + 5.5 + 3.8
= 48.4
Σy = 12.9 + 7.6 + 9 + 7.4 + 9.2 + 6.6 + 7.7 + 8.1 + 6.5 + 8.4
= 84.4
Σx^2 = (9.2)^2 + (4.3)^2 + (3.4)^2 + (8)^2 + (5.3)^2 + (5.1)^2 + (0.3)^2 + (3.3)^2 + (5.5)^2 + (3.8)^2
= 84.64 + 18.49 + 11.56 + 64 + 28.09 + 26.01 + 0.09 + 10.89 + 30.25 + 14.44
= 288.46
Σy^2 = (12.9)^2 + (7.6)^2 + (9)^2 + (7.4)^2 + (9.2)^2 + (6.6)^2 + (7.7)^2 + (8.1)^2 + (6.5)^2 + (8.4)^2
= 166.41 + 57.76 + 81 + 54.76 + 84.64 + 43.56 + 59.29 + 65.61 + 42.25 + 70.56
= 775.14
n = 10 (number of data points)
Now, let's plug these values into the formula:
r = (Σxy - (Σx * Σy) / n) / (√((Σx^2 - (Σx)^2 / n) * (Σy^2 - (Σy)^2 / n)))
= (379.19 - (48.4 * 84.4) / 10) / (√((288.46 - (48.4)^2 / 10) * (775.14 - (84.4)^2 / 10)))
= (379.19 - (48.4 * 84.4) / 10) / (√((288.46 - 234.09) * (775.14 - 711.36)))
= (379.19 - (48.4 * 84.4) / 10) / (√(54.37 * 63.78))
= (379.19 - (4089.76 / 10)) / (√(3477.0856))
= (379.19 - 408.976) / (√(3477.0856))
= -29.786 / 58.978
= -0.505 (correct to three decimal places)
Therefore, r = -0.505 (correct to three decimal places).
To calculate the critical region, we need to determine the critical value of r at α = 0.05 and a two-tailed test. The degrees of freedom (df) for the correlation coefficient is given by df = n - 2, where n is the number of data points.
In this case, df = 10 - 2 = 8.
Using a critical value table for the Pearson correlation coefficient with df = 8 at α = 0.05, we find that the critical value of r is approximately ±0.632.
Therefore, the critical region is (-∞, -0.632) U (0.632, +∞).
I hope this helps! Let me know if you have any further questions.