32 38) Forty-two percent of primary care doctors think their...

32 38) Forty-two percent of primary care doctors think their patients receive unnecessary medical care (Reader's Digest, December 2011/January 2012). Suppose a sample of 300 33 primary care doctors were taken. Answer the following: 34 a) What is the probability that the sample proportion will be within ±0.03 of the population proportion? (to 4 decimals) 35 36 37 38 39 40 41 42 p = n= $\sigma_p$ P(p?a) = P(p?b) = P(a?p?b) = #N/A #N/A #N/A #N/A 43 b) What is the probability that the sample proportion will be within ±0.05 of the population proportion? (to 4 decimals) 44 45 p = n = 46 $\sigma_p$ #N/A 47 48 P(p?a) = #N/A 49 P(p?b) = #N/A 50 P(a?p?b) = #N/A 51 52 53 a = b = #N/A #N/A #N/A b = #N/A

Transcript

00:01 For this problem, we can essentially think of this as a sampling proportion type distribution.

00:06 So if we're thinking about our sample proportions, our p -hats, they'll be distributed as a normal distribution, where the mean value is going to be a proportion equal to that of the general population, 0 .42, and the standard error of our distribution is going to be given by the square root of the population proportion 0 .42 times 1 minus the population proportion, so that would be 0 .58, divided by the sample size, in this case 300.

00:37 So we have that the standard error of our normal distribution here is roughly 0 .085.

00:45 So we have, as i said, it's roughly a normal distribution, so bell -shaped and symmetric about that mean value of 0 .42.

00:57 Now for part b, finding the probability that a sample proportion will be within plus or minus 0 .03 of the population proportion, we can use the analogy to the standard normal distribution.

01:10 And, well, since we're looking for the probability within plus or minus 0 .03, so probability of 0 .42 minus 0 .03, which would give us 0 .39 at the low end, and 0 .45 at the upper end.

01:31 Because of the symmetry of the normal distribution, this is just going to be equal to two times the probability that the sample proportion is between the population mean 0 .42 and 0 .45.

01:46 So we can do this as 2 times the probability of p hat between 0 .42 and 0 .45, which doesn't immediately seem like it's going to make our calculations any easier.

02:00 But what we can do now is use the fact that we can find the probability of p -hat being in that interval by taking the probability of the sample of proportion being less than the upper bound, then subtract off the probability of p -hat being less than the lower bound.

02:20 But because now our lower -bound is the population mean, we don't need to do any calculations to figure out that the probability of p hat being less than the population mean is simply going to be 0 .5.

02:34 It's 50 % because of the symmetry of the normal distribution.

02:38 Now to find the probability of our sample proportion being less than 0 .45, we'll translate that sample proportion value into a z score.

02:49 The corresponding z score is of course the measurement 0 .45 minus the population number.

02:56 Mean 0 .42, pardon me, divided by the standard error, roughly 0 .0285.

03:04 So we can see that the corresponding zd score is roughly 1 .05.

03:11 And of course you can use a table of values, a graph and calculator, something like that for finding the probability to the left of 0 .0.

03:19 Or, pardon me, to the left of 1 .05...

Question

Please give Ace some feedback