Question

In a random sample of 678 adults, it was determined that 58 of them have hypertension. Construct a 95% confidence interval. I'm not quite sure how to get xbar for use in the formula, and I don't know what t a/2 refers to.

Answer 1

i'm not 100% sure how to do this but i'd imagine you use the formula for the standard deviation of the binomial distribution (look on wikipedia) and then do the same thing as the previous problem

Answer 2

looks like the variance is np(1 − p) so the std dev is sqrt(np(1 − p))

Answer 3

n=678, p=58/678

Answer 4

Alright I'll try that thanks

Answer 5

do you have the answer?

Answer 6

I can look in the back of the book.

Answer 7

hold on let me guess it first

Answer 8

plus or minus 14.3 i think? so 43.7 to 72.3

Answer 9

According to this it's .065 to .107.

Answer 10

oh, they want percentages... hold on let me convert

Answer 11

BANG! that's what i got, 0.0644 to 0.1066

Answer 12

Awesome :D Thanks.

Answer 13

So, uhh, what is variance? Does that have to do with degrees of freedom? And standard deviation is just the square root of variance?

Answer 14

standard deviation is just the sq root of the variance yes

Answer 15

Okay, so I got the variance and standard deviation and everything, how can I get xbar for use in xbar +- z * stdev/sqrt n?

Answer 16

actually in this case the formula for std dev already has the sample size built into it. so you just do 1.96 * stddev (plus or minus). where std dev is sqrt(n p (1-p))

Answer 17

xbar is just 58/678

Answer 18

Isn't xbar the mean? How can it be so low?

Answer 19

And where did you get 1.96?

Answer 20

xbar is the mean probability of finding someone with hypertension

Answer 21

1.96 is the number of std devs up and down to get to 95% confidence; it's in your quick reference chart

Answer 22

Ah, okay.

Answer 23

Uhh, okay, so I have xbar = .085 +- 1.96 * (7.26/sqrt20) and I get .085 +- 3.175, that's not right...

Answer 24

n = 678
p = 58/678
variance = np (1-p) = 17980/339 = 53.04
stddev = sqrt(variance) = 7.28
95% conf interval = 1.95 * stddev = 14.27 * 1.96 = 14.27

so 95% conf is 58+14.27 (72.27) to 58-14.27 (43.72)
converting that back to probabilities you get
72.27/678 = 0.106 to
43.72/678 = 0.0644

Answer 25

O_o Sure, let me think.

Answer 26

Where did the 1.95 come from? All I see in the formula is 1.96 for Z.

Answer 27

typo it should have said 1.96

Answer 28

Oh, right.

Answer 29

And I thought xbar was 58/678?

Answer 30

it is

Answer 31

But you said 58 up there, not .085.

Answer 32

the mean is 58/678 yes. but for this particular problem i used the formula for the binomial distribution, which has built into it the number of samples. the std dev of the binomial distribution tells you how many successes you can expect with that particular number of samples (678). the mean is 58, the std dev ends up being 7. in the end you can convert that back to the same units as the sample mean by dividing the number of samples back out.

Answer 33

Oh, okay... I think I'm going to have a hard time remebering this.

Answer 34

there's probably some other way to do it in the units of percentages all the way through, i just don't happen to know it

Answer 35

Well, I got the answer, that's what matters. Thanks.

Answer 36

My god it's almost midnight and I've done half of these problems.

Answer 37

I'm gonna go get some food real quick.