Question

From a random sample of 16 bags of chips, sample mean weight is 500 grams and sample standard deviation is 3 grams. Assume that the population distribution is approximately normal. Answer the following questions 1 and 2.

1. Construct a 95% confidence interval to estimate the population mean weight. (i) State the assumptions, (ii) show your work and (iii) interpret the result in context of the problem.

2.  Suppose that you decide to collect a bigger sample to be more accurate. You want to be 99% confident that your sample mean is within 1 gram of the true mean. What is the sample size required? Use the sample standard deviation in Problem III description as an estimate for σ.

Answer 1

A random sample of 16 bags of chips, sample mean weight is 500 grams and sample standard deviation is 3 grams .

Thus   n = 16   , = 500   , s = 3

1. Construct a 95% confidence interval to estimate the population mean weight

(i) State the assumptions

The populations are normally distributed.   ( { We have already asumed it }

Each value is sampled independently from each other value

The data must be sampled randomly

Since Sample size is n = 16 < 30 we need to use t-value instead of z-value

ii)

Now 95% confidence interval to estimate the population mean weight is given by

CI = { - *    ,   + *      }

Here is t-distributed with n-1 = 15 degree of freedom and =0.05 ,

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> qt(1-0.05/2,df=15)
[1] 2.13145

Thus = 2.13145

Now = = 0.75

Thus Margin of Error ME = * = 2.13145 * 0.75 = 1.598588

So 95% confidence interval is given by

CI = { - *    ,   + *      }

   = { 500 - 2.13145 * 0.75   ,   500 + 2.13145 * 0.75 }

     = { 498.4014 , 501.5986 }

95% confidence interval to estimate the population mean weight is { 498.4014 , 501.5986 }

(iii) interpret

by 95% confidence interval , we say that we are 95% sure that population mean weight of bags of chips will be between 498.4014 Grams to   501.5986 Grams.

2.  Suppose that you decide to collect a bigger sample to be more accurate. You want to be 99% confident that your sample mean is within 1 gram of the true mean. What is the sample size required? Use the sample standard deviation in Problem III description as an estimate for σ.

Now estimate is σ is s

Thus σ = 3

Now we want to be 99% confident that your sample mean is within 1 gram of the true mean.

So we want our Marginf of Error to be ME = 1

Now ME = *

for 99% confidence z-critcal value is = 1.96 and ME = 1

Here   ME = *

By rewriting the formula in terms of n

    n = ( * σ / ME ) 2

            = ( 1.96 * 3 / 1 ) 2 = 34.5744

Thus n = 34.5744 35

So the sample size required is 35