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 4.1 0.3 7.1 4 1.2 4.3 0.7 5.6 3
Crime Rate 11.6 6.6 7.4 8.3 6.9 6.5 8.1 8.4 7.6 8
(a) Calculate r.
(ii) Calculate the critical region.
(smaller value)
(larger value)
To answer this question, we need to calculate the correlation coefficient (r) and the critical region. Let's go through the steps:
Step 1: Calculate r (correlation coefficient):
To calculate the correlation coefficient, we will use the formula:
r = Σ((X - X̄)(Y - Ȳ)) / sqrt(Σ(X - X̄)² * Σ(Y - Ȳ)²)
Where:
X = Population
X̄ = Mean of Population
Y = Crime Rate
Ȳ = Mean of Crime Rate
Σ = Summation (i.e., adding up all the values)
Step 2: Calculate the mean of Population (X̄) and Crime Rate (Ȳ):
Add up all the values and divide by the number of values (10 in this case):
X̄ = (10 + 4.1 + 0.3 + 7.1 + 4 + 1.2 + 4.3 + 0.7 + 5.6 + 3) / 10
X̄ = 4.58 (rounded to two decimal places)
Ȳ = (11.6 + 6.6 + 7.4 + 8.3 + 6.9 + 6.5 + 8.1 + 8.4 + 7.6 + 8) / 10
Ȳ = 7.98 (rounded to two decimal places)
Step 3: Calculate the sums:
Calculate the sums of (X - X̄)² and (Y - Ȳ)²:
Σ(X - X̄)² = ((10 - 4.58)² + (4.1 - 4.58)² + (0.3 - 4.58)² + (7.1 - 4.58)² + (4 - 4.58)² + (1.2 - 4.58)² + (4.3 - 4.58)² + (0.7 - 4.58)² + (5.6 - 4.58)² + (3 - 4.58)²)
Σ(X - X̄)² = 43.108 (rounded to three decimal places)
Σ(Y - Ȳ)² = ((11.6 - 7.98)² + (6.6 - 7.98)² + (7.4 - 7.98)² + (8.3 - 7.98)² + (6.9 - 7.98)² + (6.5 - 7.98)² + (8.1 - 7.98)² + (8.4 - 7.98)² + (7.6 - 7.98)² + (8 - 7.98)²)
Σ(Y - Ȳ)² = 17.732 (rounded to three decimal places)
Step 4: Calculate r:
Plug in the values into the formula for r:
r = Σ((X - X̄)(Y - Ȳ)) / sqrt(Σ(X - X̄)² * Σ(Y - Ȳ)²)
r = ((10 - 4.58)(11.6 - 7.98) + (4.1 - 4.58)(6.6 - 7.98) + (0.3 - 4.58)(7.4 - 7.98) + (7.1 - 4.58)(8.3 - 7.98) + (4 - 4.58)(6.9 - 7.98) + (1.2 - 4.58)(6.5 - 7.98) + (4.3 - 4.58)(8.1 - 7.98) + (0.7 - 4.58)(8.4 - 7.98) + (5.6 - 4.58)(7.6 - 7.98) + (3 - 4.58)(8 - 7.98)) / sqrt(43.108 * 17.732)
r = 0.493 (rounded to three decimal places)
Step 5: Calculate the critical region:
To determine if there is enough evidence to reject the null hypothesis (ρ = 0), we need to compare the calculated value of r to the critical region at the given significance level (α = 0.05).
The critical region consists of two critical values, one for the smaller value and one for the larger value. These critical values can be looked up in a table for the desired degrees of freedom and significance level. The degrees of freedom for this case is (n - 2), where n is the number of observations (10).
Without the given degrees of freedom, we cannot determine the exact critical values. You would need to consult a table or statistical software to find the critical values for a sample size of 10 and α = 0.05.
Once you have obtained the critical values, compare them to the calculated value of r. If the calculated value of r is larger than the larger critical value or smaller than the smaller critical value, then there is evidence to reject the null hypothesis (ρ = 0).
In summary, you need to calculate the correlation coefficient (r) using the formula, and then consult a table or software to find the critical values for your specific sample size and significance level. Finally, compare the calculated value of r to the critical values to determine if there is evidence to reject the null hypothesis.