Given a normal distribution with µ = 51 and ? = 4, complete...

Given a normal distribution with µ = 51 and ? = 4, complete parts (a) through (d). Click here to view page 1 of the cumulative standardized normal distribution table. Click here to view page 2 of the cumulative standardized normal distribution table. a. What is the probability that X > 45? P(X > 45) = (Round to four decimal places as needed.) b. What is the probability that X < 43? P(X < 43) = (Round to four decimal places as needed.) c. For this distribution, 9% of the values are less than what X-value? X = (Round to the nearest integer as needed.) d. Between what two X-values (symmetrically distributed around the mean) are 70% of the values? For this distribution, 70% of the values are between X = and X = (Round to the nearest integer as needed.)

Transcript

00:01 All right, so we know that the mean of our distribution is 51, and we have a standard deviation of three.

00:07 We want, for part a, what's the probability of getting greater than 47? so that's going to be, we have to find the z score for 47, and then we want to find the area above the curve, and we have the percentile table here to reference.

00:27 So the z score is going to be the raw score minus the mean over the standard deviation.

00:33 That's negative four thirds or negative 1 .3 repeating.

00:39 So then we can look up the z score.

00:43 And by symmetry, since we only have the table for positive z scores, what we know is if this is z equals negative 1 .3, and we have z equals positive 1 .3, which we can get the value of, the area under the curve, like this, this will be one minus the red area.

01:20 And this will also be one minus the red area.

01:25 What we're looking for is the area above the negative 1 .3.

01:28 So this one right here, which is going to be symmetrical to the red area.

01:40 So if we look up the z score for positive 1 .3 and find the percentile, then that is our answer for greater than negative 1 .3.

01:47 So for positive, so 1 .3.

01:52 Right here, 0 .9032.

01:55 So that is our answer.

02:00 For part b, what's the probability of being less than 48? now, what's special about 48 is if you were to calculate the z score, 48 minus 51 divided by 3, is negative 1.

02:18 It's a whole number.

02:19 And we can use the empirical rule therefore to figure that out.

02:22 So if this is z equals 0, and then we have negative 1, we know that this above 0 is 50%, and then this is half of 68 by the empirical rule.

02:34 This is 34.

02:35 So there's a total of 84%, which as a decimal is 0 .84.

02:40 You could have added up based on the percentiles from the z score table, but because it's a whole number, it's easier to just use the empirical rule and add from that.

02:53 All right.

02:54 10 % of the values are less than, oh, wait, i did greater than.

03:02 This is less than.

03:05 So it's ignore that.

03:13 So, okay, if this is 84%, then the red area is 100 minus 84, which is 16%.

03:19 So 0 .16.

03:21 That would be for less than.

03:24 Okay.

03:25 For part c, what's the 10 % of the values are less than what value? so if we're looking at only 10 % are less, it's right here.

03:41 It's going to be the 10th percentile.

03:47 So if we find the 90th, the areas will be symmetrical above and below that.

03:54 Okay, so for the 90th percentile, 0 .9, we want to find what's the z score.

04:00 Closest one is right here.

04:02 So it's 0 .128.

04:03 So if z equals, i mean, 1 .28...

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