Question

Q1. (10 points) Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men and weigh them. We find that the average in our sample weighs 180 pounds, and the standard deviation of the sample is 35 pounds. a) Construct a 99% confidence interval for the population mean. b) Interpret your conclusion

Answer 1

Answer:

a) The 99% confidence interval would be given by (177.144;182.856)

b) We are 99% confident that the true mean for the weight of an adult male in Dekalb County, Georgia is between 177.144 and 182.856.

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=180$ represent the sample mean

$\mu$ population mean (variable of interest)

s=35 represent the sample standard deviation

n=1000 represent the sample size  

Part a: Confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=1000-1=999$

Since the Confidence is 0.99 or 99%, the value of $\alpha=0.01$ and $\alpha/2 =0.005$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,999)".And we see that $t_{\alpha/2}=2.58$

Now we have everything in order to replace into formula (1):

$180-2.58\frac{35}{\sqrt{1000}}=177.144$    

$180+2.58\frac{35}{\sqrt{1000}}=182.856$    

So on this case the 99% confidence interval would be given by (177.144;182.856)    

Part b

We are 99% confident that the true mean for the weight of an adult male in Dekalb County, Georgia is between 177.144 and 182.856.