Question

Construct the confidence interval for the population mean mu. c = 0.90​, x = 16.9​, s = 9.0​, and n = 45. A 90​% confidence interval for mu is:______.

Answer 1

Answer:

The  90%  confidence interval for population mean is   $14.7 < \mu < 19.1$

Step-by-step explanation:

From the question we are told that

   The sample mean is  $\= x = 16.9$

    The confidence level is  $C = 0.90$

     The sample size is  $n = 45$

     The standard deviation

Now given that the confidence level is  0.90 the  level of significance is mathematically evaluated as

       $\alpha = 1-0.90$

       $\alpha = 0.10$

Next we obtain the critical value of  $\frac{\alpha }{2}$  from the standardized normal distribution table. The values is  $Z_{\frac{\alpha }{2} } = 1.645$

The  reason we are obtaining critical values for $\frac{\alpha }{2}$  instead of  that of  $\alpha$  is because $\alpha$  represents the area under the normal curve where the confidence level 1 - $\alpha$ (90%)  did not cover which include both the left and right tail while $\frac{\alpha }{2}$  is just considering the area of one tail which is what we required calculate the margin of error

  Generally the margin of error is mathematically evaluated as

        $MOE = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }$

substituting values

         $MOE = 1.645* \frac{ 9 }{\sqrt{45} }$

         $MOE = 2.207$

The  90%  confidence level interval is mathematically represented as

      $\= x - MOE < \mu < \= x + MOE$

substituting values

     $16.9 - 2.207 < \mu < 16.9 + 2.207$

    $16.9 - 2.207 < \mu < 16.9 + 2.207$

     $14.7 < \mu < 19.1$