00:01
We have a brand new chocolate bar being market tested, and 500 of the new chocolate bars were given to consumers, so therefore our sample size is 500, and the consumers were asked whether they liked or they disliked the chocolate bars.
00:20
And it was found that 350 liked the chocolate bars, and 150 disliked the chocolate bars.
00:31
So in part a, we want the point estimate for the proportion of people who liked the chocolate bar.
00:38
So your point estimate is p hat, and it's found by doing favorable over possible.
00:45
So there were 350 out of the possible 500 people that liked the chocolate bar.
00:53
If we were to simplify this, we would get 7 over 10 or 0 .7.
01:03
For part b, we want to calculate the 95 % confidence interval, and in order to do so, we're first going to have to find our margin of error.
01:19
And to find our margin of error, it will require a critical z score, and then we'll multiply that by the square root of p -hat times the quantity 1 minus p -hat divided by the sample size.
01:37
So in order to find the critical z, we're going to think of our bell -shaped curve.
01:45
And since we are looking for the 95 % confidence interval, we are referencing the middle 95 % of this curve.
01:54
And if 95 % is in the middle of the curve, that means there's 5 % to be split evenly between the two tails.
02:02
So each tail has an area of 0 .025.
02:07
And the critical z is going to be the boundaries of that 95 % interval.
02:14
And the most efficient way to find the first critical z is to use your inverse norm function on a graph and calculator.
02:24
And when you use that function, you need to provide the area that's in the left tail or to the left of that location, followed by the mean and the standard deviation.
02:34
So for our scenario, we'll use inverse norm.
02:39
The area in our left tail is 0 .025.
02:43
And since this is a z score, the mean of z scores is always zero, and the standard deviation is 1.
02:49
So i'm going to bring in my graphing calculator, and to find inverse norm, you're going to hit the second button and the vers button.
02:56
And this is a texas instrument brand calculator, and it happens to be number three in my menu.
03:02
So i'm going to provide the area in the left.
03:04
Left tail, followed by the mean and the standard deviation.
03:08
And i'm getting that the left critical z is a negative 1 .9599 -6 -3 -984 -5.
03:22
And because of the symmetric nature of this bell -shaped curve, which is centered around a z score equal to zero, the right critical z will be the positive version of the left one, so it's going to be positive 1 .9599639845.
03:43
So we're ready to use our margin of error formula.
03:46
So we'll say that the error equals plus or minus 1 .95996399 -845 multiplied by the square root of our p hat, which was 0 .7 times 1 minus 0 .7, divide by our sample size, which was 500.
04:11
So our margin of error turns out to be plus or minus 0 .040 1673089.
04:24
So the final part of finding our confidence interval is to take the p hat and subpoenaed...