# Determine the approximate z-scores for the dirst quartile and the third quartile of the standard normal distribution.

Question
Normal distributions
Determine the approximate z-scores for the dirst quartile and the third quartile of the standard normal distribution.

2021-01-09
The given information "z-score for the first quartile and the third quartile of the standard normal distribution."
We know that 0.250 square unit of the area covered in the standard normal distribution to the left of the third quartile.
We have to find the z-score that has an area of 0.250 square units to the left of the z-score.
So from the table of standard normal distribution we get the z-score of the area 0.250 square unit as $$z = 0.67$$
And 0.250 square unit of the area covered in the standard normal distribution to the right of the first quartile.
We have to find the z-score that has an area of 0.250 square units to
the right of the z-score.
So from the table of standard normal distribution we get the z-score of the area 0.250 square unit as $$z = 0.67$$
Since the z-score of the first quartile lies to left of 0, thus the z-score of first quartile is $$z = -0.67$$

### Relevant Questions

a) Which of the following properties distinguishes the standard normal distribution from other normal distributions?
-The mean is located at the center of the distribution.
-The total area under the curve is equal to 1.00.
-The curve is continuous.
-The mean is 0 and the standard deviation is 1.
b) Find the probability $$\displaystyle{P}{\left({z}{<}-{0.51}\right)}$$ using the standard normal distribution.
c) Find the probability $$\displaystyle{P}{\left({z}{>}-{0.59}\right)}$$ using the standard normal distribution.
For the standard normal distribution, find the following probabilities.
(b) $$Pr(z>2.5)$$
For the standard normal distribution, find the following probabilities.
(a) $$Pr(0 \leq Z \leq 2.5)$$
The manager of the store in the preceding exercise calculated the residual for each point in the scatterplot and made a dotplot of the residuals.
The distribution of residuals is roughly Normal with a mean of $0 and standard deviation of$22.92.
The middle 95% of residuals should be between which two values? Use this information to give an interval of plausible values for the weekly sales revenue if 5 linear feet are allocated to the store's brand of men's grooming products.
Basic Computation:$$\hat{p}$$ Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(c) Suppose $$n = 48$$ and $$p= 0.15$$. Can we approximate the $$\hat{p}$$ distribution by a normal distribution? Why? What are the values of $$\mu_{hat{p}}$$ and $$\sigma_{p}$$.?
Basic Computation:$$\hat{p}$$ Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(a) Suppose $$n = 33$$ and $$p = 0.21$$. Can we approximate the $$\hat{p}$$
distribution by a normal distribution? Why? What are the values of $$\mu_{hat{p}}$$ and $$\sigma_ {\hat{p}}$$.?
Assume that the random variable Z follows standard normal distribution, calculate the following probabilities (Round to two decimal places)
a)P(z>1.9)
b)$$\displaystyle{P}{\left(−{2}\le{z}\le{1.2}\right)}$$
c)P(z>−0.2)
(b) Suppose $$n= 20$$ and $$p=0.23$$. Can we safely approximate the \hat{p} distribution by a normal distribution? Why or why not?
(a) Suppose $$n = 100$$ and $$p= 0.23$$. Can we safely approximate the \hat{p} distribution by a normal distribution? Why?
Compute $$\mu_{hat{p}}$$ and $$\sigma_{hat{p}}$$.