Question 1139536
<font face="times" color="black" size="3">
n = 1051 is the sample size


Of that sample size, x = 599 said they were worried, so 
*[Tex \Large \hat{p} = \frac{x}{n} = \frac{599}{1051} \approx 0.569933]
is a good estimate of the population proportion of people worried about having enough money for retirement.


The notation *[Tex \Large \hat{p}] is read as "p-hat". Basically it's the letter "p" but with a "hat" on top so to speak. This is to help separate it from the regular letter "p" which is the population proportion; while *[Tex \Large \hat{p}] is the sample proportion.


Let's compute the standard error which I'll call "SE" for short.
*[Tex \Large \text{SE} = \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}] see note below


*[Tex \Large \text{SE} \approx \sqrt{\frac{0.569933*(1-0.569933)}{1051}}]


*[Tex \Large \text{SE} \approx \sqrt{\frac{0.569933*0.430067}{1051}}]


*[Tex \Large \text{SE} \approx \sqrt{\frac{0.245109}{1051}}]


*[Tex \Large \text{SE} \approx 0.015264]


note: we do not use p here as we don't know the population proportion. If we knew the population proportion then we wouldn't need a confidence interval (since a confidence interval is used to estimate the population proportion). We can say that p-hat is an unbiased estimator of p.


---------------------------------------------------------------------


Now onto the margin of error, which I'll abbreviate as "ME". We'll need the z critical value. At 95% confidence, the critical z value is approximately z = 1.960; this value is found using a calculator or a table. 


I used <a href = "http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">this table</a> to find the z critical value. Scroll to the bottom of the page to locate the row that starts with Z. Then locate the column that has "95%" at the very bottom. The value just above this is 1.960. A table similar to this should be found in the appendix section of your statistics textbook.


Let's use those two values to get...


*[Tex \Large \text{ME} = z*\text{SE}]


*[Tex \Large \text{ME} \approx 1.960*0.015264]


*[Tex \Large \text{ME} \approx 0.029917]

---------------------------------------------------------------------


The margin of error is then added and subtracted from the p-hat value, as the p-hat value is the best estimate of p. The p-hat value is the center of the confidence interval. The margin of error tells us how spread out or how wide the interval is.


L = lower boundary of confidence interval


*[Tex \Large L = \hat{p} - \text{ME}]


*[Tex \Large L \approx 0.569933 - 0.029917]


*[Tex \Large L \approx 0.540016]


*[Tex \Large L \approx 0.54]


The upper boundary is nearly identical, but instead we add this time.


U = upper boundary of confidence interval


*[Tex \Large U = \hat{p} + \text{ME}]


*[Tex \Large U \approx 0.569933 + 0.029917]


*[Tex \Large U \approx 0.59985]


*[Tex \Large U \approx 0.60]


---------------------------------------------------------------------


Answer: <font color=red size=4>(0.54, 0.60)</font>
This answer is approximate rounded to two decimal places. 


Interpretation: We are 95% confident that the true proportion p is between 0.54 and 0.60, meaning that we're 95% confident that the proportion of people worried about having enough money for retirement is between 54% and 60%.
</font>