# Probability calculation using the Z score

Fill in the P(X = x) values in the table below to give a legitimate probability distribution for the discrete random variable X, whose possible values are -2, -1, 4, 5, and 6.

Value x of X P(X = x)

-2 0.24

-1 0.19

4 0.12

5

6

Let X be a random variable with the following probability distribution

Value x of X P(X = x)

-2 0.10

-1 0.35

0 0.40

1 0.05

2 0.10

Find the expectation E (X) and variance Var (X) of X.

E (x) =

Var (X) =

P (Z > -2.15) =

P (Z < 0.98) =

P ( - 0.78 < Z < 2.20) =

Let Z be a standard normal random variable, Use the calculator provided to determine the value of c such that P ( -c < Z < c)= 0.9512

Carry your intermediate computations to at least four decimal places

Let Z be a standard normal random variable, Use the calculator provided to determine the value of c such that P ( -c < Z < c)= 0.9512

Let Z be a standard normal random variable, Use the calculator provided to determine the value of c such that P ( c < Z < c-1.17)= 0.0954

The scores on a particular test are normally distributed with a mean of 130 and a standard deviation of 15 what is the minimum score needed to be in the top 20% of the scores on the test. Carry your intermediate computations to a least four decimal places , and round your answer to a least one decimal place.

Normal distribution: Word problems

Suppose that the time required to complete a 1040R tax form is normally distributed with a mean of minutes and a standard deviation of minutes. What proportion of 1040R tax forms will be completed in at most minutes? Round your answer to at least four decimal places.

t distribution

Use the calculator provided to solve the following problems.

? Consider a t distribution with degrees of freedom. Compute . Round your answer to at least three decimal places.

? Consider a t distribution with degrees of freedom. Find the value of such that . Round your answer to at least three decimal places.

P (-1.18 < t < 1.18) =

C =

Chi-square distribution

.

? Suppose that follows a chi-square distribution with degrees of freedom. Compute . Round your answer to at least three decimal places.

? Suppose again that follows a chi-square distribution with degrees of freedom. Find such that . Round your answer to at least two decimal places.

? Find the median of the chi-square distribution with degrees of freedom. Round your answer to at least two decimal places.

P (x2 < 16) =

K=

Median =

F distribution

? Consider an F distribution with numerator degrees of freedom and denominator degrees of freedom. Compute . Round your answer to at least three decimal places.

? Consider an F distribution with numerator degrees of freedom and denominator degrees of freedom. Find such that . Round your answer to at least two decimal places.

P ( F < 1.44) =

C =

Central limit theorem: Sample mean

The mean salary offered to students who are graduating from Coastal State University this year is , with a standard deviation of . A random sample of Coastal State students graduating this year has been selected. What is the probability that the mean salary offer for these students is or less?

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

https://brainmass.com/statistics/chi-squared-test/probability-calculation-using-the-z-score-243278

#### Solution Summary

The solution provides step by step method for the calculation of probability for normal, t , f, and chi square distributions. Formula for the calculation and Interpretations of the results are also included.

Statistics: Z-score, sample standard deviation, standardized score

1. For a population with µ=50 and σ=10,

A. What is the z-score for X=55, X=60, X=75, X=45, X=30 and X=35?

B. Find the X value that corresponds to each of the following z-scores, z=1.00, z=0.80, z=1.50, z= -0.50, z= -0.30 and z= -1.50.

2. Find the z-score corresponding to a score of X=60 for each of the following distributions.

A. µ=50 and σ=10

B. µ=50 and σ=5

C. µ=70 and σ=20

D. µ=70 and σ=5

3. For a sample with a mean of M=85, a score of X=90 corresponds to z=0.50. What is the sample standard deviation?

4. In a population of exam scores, a score of X=88 corresponds to z=+2.00 and a score of X=79 corresponds to z= -1.00. What is the means for the population? What is the standard deviation for the population?

5. A distribution with a means of µ=38 and a standard deviation of σ=20 is being transformed into a standardized distribution with µ=50 and σ=10. Find the new, standardized score for each of the following values from the original population.

A. X=48

B. X=40

C. X=30

D. X=18

PLEASE SHOW STEP BY STEP HOW YOU SOLVE THESE PROBLEMS.

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