Question

Given a normal distribution with mean equals 54 and st. dev. equals3​, complete parts​ (a) through​ (d). Click here to view page 1 of the cumulative standardized normal distribution table.LOADING...  Click here to view page 2 of the cumulative standardized normal distribution table.LOADING... a. What is the probability that Xgreater than49​? ​P(Xgreater than49​)equals nothing ​(Round to four decimal places as​ needed.) b. What is the probability that Xless than51​? ​P(Xless than51​)equals nothing ​(Round to four decimal places as​ needed.) c. For this​ distribution, 5​% of the values are less than what​ X-value? Xequals nothing ​(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 Xequals nothing and Xequals nothing.

Answer 1

We are given the distribution here as:

a) The probability here is computed as:

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.9522 is the required probability here.

b) The probability here is computed as:

P(X < 51)

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.1587 is the required probability here.

c) From standard normal tables, we have:
P(Z < -1.645) = 0.05

Therefore the X value here is computed as:

= Mean - 1.645*Std Dev

= 54 - 1.645*3

= 49.065

Therefore 49.065 is the required X value here.

d) From standard normal tables, we have here:

P(Z < 1.036) = 0.85

Therefore due to symmetry, we get here:
P( -1.036 < Z < 1.036) = 0.7

Therefore the X values here are computed as:
= Mean - 1.036*Std Dev, Mean + 1.036*Std Dev

= 54 - 1.036*3, 54 + 3*1.036

= 50.892, 57.108

These are the required values here.