Question

Assume that z-scores are normally distributed with a mean of 0 and a standard deviation of 1. If P ( − 0.7 < z < b ) = 0.749 , find b. b= (round to three decimal places)

Answer 1

here from normal distribution table:

P(-0.7 <z <b) =0.749

P(Z<b)-P(Z<-0.7)=0.749

P(Z<b)-0.2420=0.7490

P(Z<b)=0.9910

b=2.366

Answer 2

To find the value of "b," we can use the properties of the standard normal distribution and the given probability.

For a standard normal distribution, the cumulative distribution function (CDF) gives the probability that a standard normal random variable is less than or equal to a specific value. Since z-scores are normally distributed with a mean of 0 and a standard deviation of 1, we can use the standard normal CDF to find the probability that z lies between −0.7 and "b."

Using a standard normal table or a calculator, we find that the CDF at z = −0.7 is approximately 0.242.

Now, the probability that −0.7 < z < b is given as 0.749. We can find b using the following equation:

P(−0.7 < z < b) = P(z < b) − P(z < −0.7) = 0.749

Substituting the CDF values:

P(z < b) − 0.242 = 0.749

Now, let's solve for b:

P(z < b) = 0.749 + 0.242 P(z < b) = 0.991

Using a standard normal table or calculator, the z-score corresponding to a cumulative probability of 0.991 is approximately 2.33.

Therefore, the value of "b" is approximately 2.33 (rounded to three decimal places).