Question

1. fbi crime rates for a random sample of ( n_1 = 9 ) in the southern region have a violent crime rates of ( m_1 = 3.75 ) with ( ss_1 = 6.9 ). in the rocky mountain states a random sample of ( n_2 = 10 ) has ( m_2 = 3.87 ) with ( ss_2 = 8.0 ). use ( alpha = 0.05 ) to determine if the crime rate in the midwestern states is significantly different.

a) identify the null and alternate hypotheses
b) compute the degrees of freedom
c) identify the critical ( t ) value
d) calculate the pooled variance

Explanation:

Step1: Identify hypotheses

Null hypothesis \(H_0:\mu_1 = \mu_2\). Alternate hypothesis \(H_1:\mu_1
eq\mu_2\).

Step2: Compute degrees of freedom

Degrees of freedom \(df=n_1 + n_2-2=9 + 10-2 = 17\).

Step3: Identify critical t - value

For \(\alpha = 0.05\) (two - tailed) and \(df = 17\), from t - distribution table, critical \(t=\pm 2.110\).

Step4: Calculate pooled variance

Pooled variance \(s_p^2=\frac{SS_1+SS_2}{n_1 + n_2-2}=\frac{6.9 + 8.0}{9+10 - 2}=\frac{14.9}{17}\approx0.876\).

Answer:

a) \(H_0:\mu_1=\mu_2\), \(H_1:\mu_1
eq\mu_2\)
b) \(17\)
c) \(\pm2.110\)
d) \(0.876\)