Question 1074068
A random sample of 45 home theater systems has a mean price of ​$140.00
Assume the population standard deviation is ​$19.40


population standard deviation is 19.40
sample size is 45.
standard error of the sample mean is 19.40 / sqrt(45) = 2.89198


at 90% confidence level, critical z is plus or minus 1.645.


at 95% confidence level, critical z is plus or minus 1.96.


critical raw score will be sample mean plus or minus critical z-score * standard error of the mean.


your sample mean is 140.
your standard error of the mean is 2.89198


at 90% confidence level, your critical raw score will be 140 plus or minus 1.645 * 2.89198.


that becomes 140 plus or minus 4.76 rounded to the nearest penny.


at 90% confidence level, the true population mean will be between 135.24 and 144.76.


at 95% confidence level, your critical raw score will be 140 plus or minus 1.96 * 2.89198.


that becomes 140 plus or minus 5.67 rounded to the nearest penny.


at 95% confidence level, the true population mean will be between 134.33 and 145.67.


the following statistics calculator supports these conclusions with an associated picture of the distribution curve.


<a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>


the first 2 graphs show the critical z-scores.


the second 2 graphs show the critical raw scores.


<img src = "http://theo.x10hosting.com/2017/032501.jpg" alt="$$$" </>


<img src = "http://theo.x10hosting.com/2017/032502.jpg" alt="$$$" </>


<img src = "http://theo.x10hosting.com/2017/032503.jpg" alt="$$$" </>


<img src = "http://theo.x10hosting.com/2017/032504.jpg" alt="$$$" </>


when you're looking for z-scores, the mean is 0 and the standard deviation is 1.


for these results, when you're looking for raw scores, the mean is 140 and the standard deviation is 1.89198 which is the standard error we calculated earlier.


as you can see, when the confidence level is greater, the spread of the data will also be greater.