Given a normal distribution with ? = 100 and ? = 10,...

Given a normal distribution with ? = 100 and ? = 10, 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 > 75? The probability that X > 75 is 0.9938 . (Round to four decimal places as needed.) b. What is the probability that X < 95? The probability that X < 95 is 0.3085 . (Round to four decimal places as needed.) c. What is the probability that X < 90 or X > 115? The probability that X < 90 or X > 115 is 0.2255 . (Round to four decimal places as needed.) d. 90% of the values are between what two X-values (symmetrically distributed around the mean)? 90% of the values are greater than and less than . (Round to two decimal places as needed.)

Transcript

00:01 Have a problem on normal distribution which is given normal distribution the mean new is given as 100 and standard deviation sigma is given as 10 so we are required to solve four different parts in this problem part a you have to find probability x greater than 75 so let us solve this we have the transformation to standard normal variable variant like this zc equals x minus mu over sigma which will use here here zc negative will come as we take x as 75 so 75 minus 100 over 10 sigma is 10 so that comes as minus 2 .5 so that comes as minus 2 .5.

01:33 So we have the z critical value as minus 2 .5.

01:41 Now probability z critical value equals minus 2 .5 that comes as 0 .0062 from negative z chart.

02:10 If you refer to the z chart, we'll get this.

02:16 Therefore probability x greater than 75 will be 1 minus 0 .0062 and that comes as 0 .9938 so this is our part now let us move to part b before moving on to part b just a brief explanation on the above result if we plot schematic of the normal curve or off sketch so this is zc negative which in this case comes as minus 2 .5 this is this point and from the negative z chart we are given this area which is 0 .0062 so now let me use a different color so we are required to find the area on the right -hand side of zc.

03:46 So this is the area we have to find, which is obviously 1 minus 0 .0062 and that is 0 .9938 as the total area under the curve is 1.

04:03 So this explains the part a and similarly the other parts will have in the same explanation.

04:09 So now let me move to part b.

04:16 Part b, we have to find probability of x less than 95.

04:24 Now how do we solve this? we will apply the same methodology.

04:33 Z critical value negative will come here as 95 minus mu is 100 and sigma over sigma that is 10 which comes as minus 0 .5.

04:49 So probability of this critical value equal to minus 0 .5 is 0 .3085.

05:02 This is again from negative z chart.

05:13 Therefore, using the same explanation or same reasoning as in part a, we can say probability x less than 95 equals 0 .3085.

05:33 I'll just show you this is zc negative and this is minus 0 .5.

05:56 And from the negative chart, from the negative chart, we have this area given as 0 .3085.

06:12 So we are required to find this probability only.

06:17 Therefore this is our answer.

06:21 So now let us move to part c.

06:28 Part c, we are asked to find probability x less than 90 or probability x greater than 115.

06:54 So how do you solve this? so let us find the z critical value in both the cases.

07:02 So zc negative equals 90 minus 100 over 10 and that is equal to minus 1 .1.

07:12 And zc positive equals 115 minus 100 over 10 and that is equal to 1 .5.

07:30 So these are the two critical values that we get.

07:35 Now probability zc negative equals minus 1 .0 is given as 0 .1587 probability zc negative equals minus 1 .0 is given as 0 .1587 from negative z chart.

08:02 And probability zc plus equal to 1 .5 is given as 0 .9332 from positive z chart.

08:28 Therefore probability x less than 90 and that union probability x greater than 115.

08:45 Is equal to 0 .1587 plus and 0 .9332 is on the left side.

08:57 So 1 minus the total area is 1 .1 .9332 which is 0 .1587 plus 0 .0 .06688 and this comes as 0 .0 .0668 and this comes as 0 .2 to 5257 plus 0 .06668 and this comes as 0 .2 to 5.

09:20 So this is our answer for part c.

09:24 Now let us move to part d.

09:28 In part d, we have to find the range of values corresponding to 90 % probability centered around the mean.

10:10 So how can we solve it? so if we draw the curve once more, this is zc positive and this is zc next.

10:42 Negative and this is the rejection region.

10:49 So let me change the color.

10:54 This is the acceptance region which is 90%.

11:01 So this is total.

11:04 This is total test.

11:08 So this will correspond to 0 .05 and this will also correspond to 0 .05 because the remaining area is 10%.

11:19 We have to select this range of the scores or the marks or values or whatever it is from here to here.

11:31 So this is our range.

11:33 So now actually we have to find, sorry, so now we have to find the z critical value.

11:47 We can find any one of the critical values, whether it is negative or positive...

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