Question

A nutrition laboratory tested 25 "reduced sodium" hotdogs of a certain brand, finding that the mean sodium content is 310 mg with a standard deviation of 36 mg.
Construct a 95% confidence interval for the mean sodium content of this brand of hot dog and interpret a 95% level of confidence. Show all work

Answer 1

Answer:

The  95% confidence interval is  $295.9 < \mu< 324.1$

A   95% level of confidence mean that there is 95%  chance  that the true population mean will be in this interval

Step-by-step explanation:

From the question we are told that

    The sample size is  $n = 25$

    The mean is  $\= x = 310 \ mg$

     The standard deviation is  $\sigma = 36 \ mg$

Given that the confidence level is  95% then the level of significance is mathematically represented as

           $\alpha = 100 - 95$

=>        $\alpha = 5\%$

=>        $\alpha = 0.05$

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

           $Z_{\frac{\alpha }{2} } =Z_{\frac{0.05 }{2} } = 1.96$

Generally the margin of error is mathematically represented as

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

substituting values

        $E = 1.96 * \frac{36 }{\sqrt{25} }$

        $E = 14.1$

The 95% level of confidence interval  is mathematically represented as

      $\= x - E < \mu$

substituting values

     $310- 14.1 < \mu< 310+ 14.1$

     $295.9 < \mu< 324.1$

The  95% level of confidence mean that there is 95%  chance  that the true population mean will be in this interval