Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. The mean is found to be 13.6 reproductions and the population standard deviation is known to be 1.9. If a sample of 189 was used for the study, construct the 85% confidence interval for the true mean number of reproductions per hour for the bacteria. Round your answers to one decimal place.

Answer 1

Answer:

The confidence interval is  $13.4< \mu < 13.8$

Step-by-step explanation:

From the question we are told that

   The sample  mean is $\= x = 13.6$

     The standard deviation is $\sigma = 1.9$

     The sample  size is  $n = 189$

given that the confidence level is  85% then the level of significance is mathematically represented as

            $\alpha = (100 - 85 )\%$

            $\alpha = 0.15$

Next we obtain the critical value  of $\frac{\alpha }{2}$ from the normal distribution table  the value  is  

      $Z_{\frac{\alpha }{2} } = 1.44$

Generally the margin of error is mathematically represented as

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

=>     $E = 1.44* \frac{1.9}{\sqrt{189} }$

=>     $E = 0.1990$

The  85%  confidence interval is mathematically represented as  

        $\= x - E < \mu <\= x + E$

=>      $13.6- 0.1990 < \mu < 13.6+ 0.1990$

=>      $13.4< \mu < 13.8$