7.1 Cover confidence intervals. Here are things to cover:
confidence interval, confidence level
estimate $\pm$ margin of error = (p - a, p+a)
We need three things to make this
A good point estimator for the parameter (ybar for $\mu$)
The sampling distribution for the estimator
the desired level of confidence
The first example is sample proportion.
A CUNY student survey was taken of 11 students to see if they know that today is teach-cuny day. 4 say yes. based on this, what is an estimate for the true proportion? Give a 75% C.I. and a 95% C.I.
Along the way, define standard error
The standard error $SE(p)$ is the standard deviation of the estimator after all the parameters are estimated. For us
$$SE(p) = \sqrt{\frac{p(1-p)}{n}}$$
Manipulate.
100 CUNY Professors were surveyed at random, they were asked if they were concerned that library funding has been reduced from 15 to 10 to 8.6 million dollars over the years 1990, 1995, 2000. 65% said, yes. Give a 75% and 95% confidence interval for the true proportion.
A study is being designed to test if students are happy with the adjunvt teaching ratio of 56% at CUNY schools. They would like to find a margin of error no more than .03 and a 95% confidence level. How many students should they survey?
It is supposed that the number of faculty at CUNY should be approximately normal with mean unknown and std 300. 4 years of data are collected and the numbers are 6886, 6515, 5244, 5594. Based on this, give a 75% and 95% CI for the mean.
$$xbar=6059.75, s=768$$
Work out these formula
$$\bar{y} \pm z^*\sigma/\sqrt{n}, B = z^* \sigma/\sqrt{n}, n = (z^* \sigma/B)^2$$
A medical researcher is investigation gestation periods for women. They believe that the std. is 1 week, but are unsure about the mean. They study 100 women and find an average of 273.5. What is a 95% CI for the true mean (assume it is normally dist.)
Repeat the salary one, only now assume you don’t know $\sigma$. Now what to do?
Mention
$$t = \frac{\bar{x} - \mu}{s/\sqrt{n}},$$
$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}, SE(\bar{x}) = \frac{s}{\sqrt{n}}$$
Compare the $t$ score at various values: 1,2,3