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.
0.444
(ii) Calculate the critical region.
(smaller value)
(larger value)
What are you test ?
Population and crime rate?
Sorry the r value..
To calculate the correlation coefficient (r), you can use the formula:
r = (nΣxy - ΣxΣy) / sqrt((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
where:
- n is the number of data points,
- Σxy is the sum of the product of the x and y values,
- Σx is the sum of the x values,
- Σy is the sum of the y values,
- Σx^2 is the sum of the squared x values,
- Σy^2 is the sum of the squared y values.
In this case, you have the following 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
1. Calculate the products of x and y:
9.2 * 12.9 = 118.68
4.3 * 7.6 = 32.68
3.4 * 9 = 30.6
8 * 7.4 = 59.2
5.3 * 9.2 = 48.76
5.1 * 6.6 = 33.66
0.3 * 7.7 = 2.31
3.3 * 8.1 = 26.73
5.5 * 6.5 = 35.75
3.8 * 8.4 = 31.92
2. Calculate the sums:
Σxy = 118.68 + 32.68 + 30.6 + 59.2 + 48.76 + 33.66 + 2.31 + 26.73 + 35.75 + 31.92 = 399.29
Σx = 9.2 + 4.3 + 3.4 + 8 + 5.3 + 5.1 + 0.3 + 3.3 + 5.5 + 3.8 = 48.9
Σy = 12.9 + 7.6 + 9 + 7.4 + 9.2 + 6.6 + 7.7 + 8.1 + 6.5 + 8.4 = 83.4
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 = 290.46
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 = 763.3
3. Plug the values into the formula:
r = (10 * 399.29 - 48.9 * 83.4) / sqrt((10 * 290.46 - (48.9)^2)(10 * 763.3 - (83.4)^2))
Calculating this formula, you get:
r ≈ 0.444
Now, to determine the critical region, you need to look at the critical value of r for a significance level of α = 0.05 (or 5%).
The critical value depends on the sample size (n) and the significance level. In this case, with n = 10, you can use a table or a statistical calculator to find the critical value for a two-tailed test.
From the table or calculator, the critical value of r at α = 0.05 and with 10 data points is around ±0.632.
Therefore, the critical region consists of correlation coefficients less than -0.632 and greater than +0.632.
So, to determine if the data provides evidence to reject the null hypothesis that ρ = 0 in favor of ρ ≠ 0 at α = 0.05, you need to compare the calculated r value (0.444) with the critical region (-0.632 to +0.632). If the calculated r falls outside the critical region, then you can reject the null hypothesis. If it falls within the critical region, then you do not have enough evidence to reject the null hypothesis.