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Video transcript

a set of laptop prices are normally distributed with the mean of $750 and a standard deviation of $60.00 what proportion of laptop prices are between 624 dollars and seven hundred and sixty eight dollars so let's think about what they are asking so we have a normal distribution for the prices so it would look something like this and this is just my hand-drawn sketch of a normal distribution so it looks something like this it should be symmetric so I'm making it as symmetric as I can hand draw it and we have the mean right in the center so the mean would be right there and that is 750 dollars they also tell us that we have a standard deviation of $60 so that means one standard deviation above the mean would be roughly right over here and that'd be 750 plus 60 so that would be eight hundred and ten dollars one standard deviation below the mean would put us right about there and that would be 750 - $60 which would be six hundred and ninety dollars and then they tell us what proportion of laptop prices are between six hundred twenty four dollars and seven hundred and sixty eight dollars so the lower bound six hundred and twenty-four dollars that's going to actually be more than another standard deviation less so that's going to be right around here so that is six hundred and twenty four dollars and seven hundred sixty-eight would put us right at about right at about there once again this is just a hand-drawn sketch but that is seven hundred sixty eight and so what proportion are between those two values so we want to find essentially the area under this distribution between these two values the way we are going to approach it we're going to figure out the z-score for 768 it's going to be positive because it's above the mean and then we're going to use a z table to figure out what proportion is below was 768 so we're essentially going to figure out this entire area we're even going to figure out the stuff that's below 624 that's what that Z table will give us then we'll figure out the z-score for 624 that will be negative two point something and we will use the Z table again to figure out the proportion that is less than that and so then we can subtract this red area from the proportion that is less than 768 to get this area in between so let's do that let's figure out first the z-score for 768 and then we'll do it for 624 the z-score for 768 I'll write it like that is going to be 768 - 750 over the standard deviation over 60 so this is going to be equal to 18 over 60 which is the same thing as 6 over let's see if we divide the numerator denominator by 3 620 it's and this is the same thing as zero point three zero so that is the z score for this upper bound let's figure out what proportion is less than that for that we take out as you table get our Z table and let's see we want to get zero point three zero and so this is zero point three this first column and we've done this in other videos this goes up until the tenths place for our z-score and then if we want to go to our hundredths place that's what these other columns give us but we're at zero point three so we're going to be in this row and our hundredths place is right over here it's a zero so this is the proportion that is less than seven hundred and sixty eight dollars so 0.6 179 so zero point six one zero point six one seven nine so now let's do the same exercise but do it for the proportion that's below 624 dollars the z-score for 624 is going to be equal to 624 minus the mean of 750 all of that over 60 and so what is that going to be I'll get my calculator out for this one still want to make a careless error 624 minus 750 is equal to and then divide by 60 is equal to negative two point one so that lower bound is two point one standard deviations below the mean or you could say it has a z-score of negative two point one is equal to negative two point one and so to figure out the proportion that is less than that this red area right over here we go back to our Z table and we'd actually go to the first part of the Z table so same idea but this starts at negative a z-score of negative three point four three point four standard deviations below the mean but just like we saw before this is our zero hundredths 102 hundredths so on and so forth and we want to go to negative two point one we could say negative two point one zero just to be precise so this is going to get us it's C negative two point one there we go and so we are negative two point one and it's negative two point one zero so we have zero hundredths so we're going to be right here on our table so we see the proportion that is less than 624 is 0.001 seven nine or zero point zero one seven nine so zero point zero one seven nine and so if we want to figure out the proportion that's in between the two we just subtract this red area from this entire area the entire proportion that's less than 768 to get what's in between zero point six one seven nine once again and I keep repeating it that's this entire area right over here and we're going subtract out what we have in red minus 0.017 9 so we're going to subtract this out to get 0.6 so if we want to give our answer to four decimal places it would be zero point six zero zero zero or another way to think about it is exactly 60% is between 624 and 768
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