The heights of female students attending a Midwestern...

The heights of female students attending a Midwestern college are approximately normally distributed with mean 64.4 inches and s. d. 2.4 inches. a.What percentages of female students have heights between 62 and 63 inches? b. If a student’s height is in the lowest 10%,what’s the student’s actual height? c. What is the probability that a selected student’s height is more than 65 inches? d. Extra credit: Find the range within which middle 40% of the female students heights lie? e. If a random sample of 25 female students are selected, find the probability that their average height is more than 65 inches. f. If the heights of female students are not normally distributed, what will be your answer to part(e)? Why?

Transcript

00:01 Hello students, we have given the mean of the normal distribution that is mu is equal to 64 .4.

00:07 Then the standard deviation sigma is equal to 2 .4.

00:10 Now by using this in the a part we have to calculate the percentages of the female students have heights between 62 and 63 that is probability of 62 is less than x is less than 63.

00:23 This is we need to find now here by using the normal approximation we can write this probability as 62 minus 64 .4 divide by 2 .4 is less than x minus mu divide by sigma is less than 63 minus 64 .4 divide by 2 .4 and now x minus mu by sigma is nothing but standard normal variable that is z.

00:50 So this is probability of minus 1 is less than z is less than minus 0 .583 and which is equal to probability of z less than minus 0 .583 minus probability of z less than minus 1 and by using the z table this probability 0 .2809 minus 0 .1587 that is equal to 0 .1222 which is approximately equal to 12 .22%.

01:24 Then in the next question, we have to find the value of x such that we have given probability of z less than z is equal to 0 .10.

01:36 Now by using the z table z table we get probability of z less than minus 1 .28 is equal to 0 .10 and if we compare these two expressions, then we will get the value of z is equal to minus 1 .28 and we know the formula for the z score is x minus mu divide by sigma that is equal to minus 1 .28.

02:07 Therefore x minus mu is 64 .4 divide by sigma is 2 .4 that is equal to minus 1 .28.

02:18 Therefore, we will get the value of x is equal to 61 .328.

02:24 Then in the c part, we have to calculate the probability that a selected student's height is more than 65 inches.

02:35 So probability of x greater than 65 is equal to probability of z greater than 65 minus 64 .4 divide by 2 .4 and that is equal to probability of z greater than 0 .25 is equal to 0 .4013.

02:55 Then in the d part, we have to find the range within which middle 40 % of the female students heights lie.

03:07 So suppose probability of minus x1 is less than x is less than x1 is equal to 0 .40.

03:16 Then we can write this probability as probability of x less than x1 minus probability of x less than minus x1 which is equal to 0 .40...

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