Using Confidence Intervals

Using Confidence Intervals


Today we are going to make an 80% confidence
interval for the average weight of a 4 year old. So we will know whether it will ruin your back to
have one hold on like a monkey. So, we have an average X- bar, which is 40lbs. Notice, sigma is the true standard deviation. How do we know that? We know that magically, because I’m your instructor,
and when I say that’s sigma, you just believe it. At some point we will talk about a more realistic scenario
where we maybe don’t know sigma. The first place we are going to start is by
drawing a picture. I have a normal curve like this. Centered at a mean I don’t know, but I want to make a
confident interval that captures 80% of the interval. If I want 80% in the middle, how much should be on each
side? Not 20%. 20% plus 80% plus another 20% would be more than 100%. I need half of 20% in each tail, which is
going to be 10% here and 10% there. Today we are going to make an 80% confidence
interval for the average weight of a 4 year old. So we will know whether it will ruin your back to
have one hold on like a monkey. So, we have an average X- bar, which is 40lbs. Notice, sigma is the
true standard deviation. How do we know that? We know that magically, because I’m your instructor,
and when I say that’s sigma, you just believe it. At some point we will talk about a more realistic scenario
where we maybe don’t know sigma. The first place we are going to start is by
drawing a picture. I have a normal curve like this. Centered at a mean I don’t know, but I want to make a
confident interval that captures 80% of the interval. If I want 80% in the middle, how much should be on each
side? Not 20%. 20% plus 80% plus another 20% would be more than 100%. I need half of 20% in each
tail, which is going to be 10% here and 10% there. So, 10% is what I want to put in right there,
and I get a z score of 1.282. So, I make my confidence interval as 1.282 times my sigma which is 10. N is over
30 over the square root of 40, that Z score needs to be plus minus, and my average is going to be 40. So,
our confidence interval is 37.97 to 42.03. That is our confidence interval. Can we use
this? This is how we can use this: imagine if someone came to you and said, “I think mu is 45”. You
would look at you confidence interval and say, “No way!” No way, 45 is not in that interval; I can tell
you mu isn’t 45 with 80% confidence. If someone said to you, “I think the mu is 38, you would say, “yes
way!” Because 38 is in the interval, and so it is reasonable that does not prove mu IS 38. It only says a mu
of 38 is not unreasonable. The same way as fail to reject is not the same as saying reject the null. Now,
over there we have a hypothesis that mu could be 43. What do you think of 43, yes or no? No way! We
know that our Mu can’t be 43, because our confidence interval says no. So, we would reject that
hypothesis. So, that hypothesis interval tells us a reasonable range of values for mu.

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