Question

Heights for a certain group of people are normally distributed with mean-64 inches and standard deviation-2.9 inches. Find the proportion of people in the group whose heights fall into the following ranges. (Round your answers to four decimal places.) (a) Between 61 inches and 64 inches (b) Between 57 Inches and 71 inches. (c) Less than 71 inches. (d) Greater than 57 inches (e) Either less than 57 Inches or greater than 71 inches You may need to use the appropriate table in the Appendix of Tables to answer this question. Need Help? /2 points UHStats&.E.079 My Notes Ask Your Te Heights for group of people nor mally distributed ith mean 63 inches and standard deviation 2.6 inches. Find the height h such that about 25% of people in the group are shorter than that height. Round the an er to the nearest tenth. in. what is the percentile, ρ ranking for that height?

Answer 1

1)a) P(61 < X < 64)

= P((61 - $\mu$)/$\sigma$ < (X - $\mu$)/$\sigma$ < (64 - $\mu$)/$\sigma$)

= P((61 - 64)/2.9 < Z < (64 - 64)/2.9)

= P(-1.03 < Z < 0)

= P(Z < 0) - P(Z < -1.03)

= 0.5 - 0.1515

= 0.3485

b) P(57 < X < 71)

= P((57 - $\mu$)/$\sigma$ < (X - $\mu$)/$\sigma$ < (71 - $\mu$)/$\sigma$)

= P((57 - 64)/2.9 < Z < (71 - 64)/2.9)

= P(-2.41 < Z < 2.41)

= P(Z < 2.41) - P(Z < -2.41)

= 0.9920 - 0.0080

= 0.9840

c) P(X < 71)

= P((X - $\mu$)/$\sigma$ < (71 - $\mu$)/$\sigma$)

= P(Z < (71 - 64)/2.9)

= P(Z < 2.41)

= 0.9920

d) P(X > 57)

= P((X - $\mu$)/$\sigma$ > (57 - $\mu$)/$\sigma$)

= P(Z > (57 - 64)/2.9)

= P(Z > -2.41)

= 1 - P(Z < -2.41)

= 1 - 0.0080

= 0.9920

e) P(X < 57) + P(X > 71)

= P((X - $\mu$)/$\sigma$ < (57 - $\mu$)/$\sigma$) + P((X - $\mu$)/$\sigma$ > (71 - $\mu$)/$\sigma$)

= P(Z < (57 - 64)/2.9) + P(Z > (71 - 64)/2.9)

= P(Z < -2.41) + P(Z > 2.41)

= P(Z < -2.41) + (1 - P(Z < 2.41))

= 0.0080 + (1 - 0.9920)

= 0.0080 + 0.0080 = 0.0160

2)a) P(X < h) = 0.25

or, P((X - $\mu$)/$\sigma$ < (h - $\mu$)/$\sigma$) = 0.25

or, P(Z < (h - 63)/2.6) = 0.25

or, (h - 63)/2.6 = -0.67

or, h = -0.67 * 2.6 + 63

or, h = 61.3

p = 25%