Question

The intelligence quotients (IQs) of 16 students from one area of a city showed a mean of 107 and a standard deviation of 10, while the IQs of 14 students from another area of the city showed a mean of 112 and a standard deviation of 8. Is there a significant difference between the IQs of the two groups at significance level of 0.01. What is the alternative hypothesis?

Answer 1

Answer:

• No, there is no significant difference between the IQs of the two groups.

• Alternative Hypothesis is that the two groups have different IQs os students.

Step-by-step explanation:

We are provided that IQs of 16 students from one area of a city had a mean of 107 and a standard deviation of 10 while the IQs of 14 students from another area of the city had a mean of 112 and a standard deviation of 8.

And we have to check that is there a significant difference between the IQs of the two groups.

Firstly let,    Null Hypothesis, $H_0$ : The two groups have same IQs { $\mu_1 = \mu_2$ }

          Alternate Hypothesis, $H_1$ : The two groups have different IQs{ $\mu_1 \neq \mu_2$}

Since we don't know about population standard deviations;

The test statistics we will use here will be ;

            $\frac{(X_1bar - X_2bar)- (\mu_1 - \mu_2) }{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ follows t distribution with $(n_1 + n_2 -2)$ degree

                                                    of freedom { $t_n__1 +n_2 - 2$ }

 Here, $X_1bar$ = 107      $X_2bar$ = 112         $s_1$ = 10          $s_2$ = 8

           $n_1$ = 16                $n_2$ = 14

           $s_p$ = $\sqrt{\frac{(n_1 - 1)*s_1^{2} + (n_2 -1)*s_2^{2} }{(n_1 + n_2 -2)} }$ = 9.1261

  Test statistics = $\frac{(107-112) - 0}{9.1261*\sqrt{\frac{1}{16} +\frac{1}{14} } }$  follows $t_2_8$

                           = -1.50

Now at 1% level of significance t table is giving the critical value of -2.467 and our test statistics is higher than this means it does not fall in the rejection region so we will accept our null hypothesis and conclude that there is no significant difference between the IQs of the two groups.