The Normal distribution below models the distribution of...

The Normal distribution below models the distribution of highway gas consumption (in miles per gallon) for a large collection of SUVs when driving on level roads. The centre is shown at 24.8 mpg and the standard deviations reflect the spread of the data. (a) Find the standard deviation so that the top 2.5% of SUVs get 37.2 mpg or higher. 68% 95% 99.7% 6.2 12.4 18.6 24.8 31.0 37.2 43.4 mpg (b) What percentage of SUV's consume less than 19 miles per gallon?

Transcript

00:01 We are looking at a normal distribution here, but some values have been labelled on the curve.

00:07 And the values labelled are those belonging to the empirical rule.

00:14 So when you see 68, 95, and 99 .7%, you're looking at the empirical rule.

00:21 And it's all about the percentage, roughly, of values within a certain number of standard deviations of the mean.

00:29 68 % is within one standard deviation, sigma, 95 % is within 2 standard deviations, 99 .7 % is within 3 standard deviations.

00:48 So that's what it's showing you there.

00:52 We need to know the standard deviation so that the top 2 .5 % are getting 37 .2 or higher.

01:01 So if i draw out what's happening here, right again.

01:08 So we have this curve and 37 .2 is up here.

01:18 So this bit is 2 .5%.

01:21 Okay, so how can i break up the others? so 2 .5%, i'm looking at this middle value here, 95%.

01:30 So if i put another value here, so if i say, okay, this is two standard deviations above the mean, and this is 2 below the mean, then this would also be 2 .5%, and this middle bit will be 95%.

01:44 This is because of a symmetry.

01:46 If the values are the same number of standard deviations away from the mean, the tails are the same size.

01:56 So now following this, i know that 37 .2 is two standard deviations above the mean.

02:06 Splitting up like that is very important for this part.

02:09 Now i know that 37 .5.

02:12 Is the mean, which is 24 .8, plus two standard deviations.

02:21 So if i take away 24 .7, from both, 24 .8 from both sides, i get 12 .4.

02:31 I get the standard deviation is 6 .2.

02:38 Now for part b, what percent consume less than 19 miles per gallon? okay, so 19 is not a nice round number.

02:56 That's...

02:57 So 19 minus the mean, 24 .8, is 5 .8 below the mean, not a nice round number of standard deviations...

Question

Please give Ace some feedback