ECO 578 EXAM II | Complete Solution

                                                                                                                     Name: _______________
                                                                       ID:      _______________
Online Exam II

Part A: Multiple Choice (1–10)

____1. The cumulative probability distribution of a random variable X gives the probability that X is _______ to x_0, some spacified value of X.
    Greater than or equal            c.   Less than or equal
    Equal                    d.   None of the above

_____2. The_______is the smallest level of significance at which H_o can be rejected.
    Value of α                    c.   p value
    Probability of commiting of Type I error    d.  vale of 1 – α
    
_____3. What is the probability of P(-1.4 < Z < 0.6)?
    0.9254                    c.   0.3427
    0.6449                    d.   0.9788

_____4. By using the binomial table, if the sample size is 20 and p equals to 0.70, what is the
  value for P(X=18)?
    0.0279                    c.    0.1820
    0.0375                    d.    0.1789

_____5. In a standard normal distribution, what is the area which lies between Z = -1.72 and     
       Z = 2.53?
    0.8948                    c. 0.9516
    0.9123                    d. 0.8604
_____6. A random sample of 60 items is taken producing a sample mean of 25 and a sample standard deviation of 12.25. What is the value for 95% confidence interval to estimate the population mean?
    23.3844≤μ≤24.8966            c.   28.3541≤μ≤29.1359
    24.1144≤μ≤25.8856            d.   25.8252≤μ≤26.5478

_____7. You perform a hypothesis test about a population mean on the basis of the following information: the sampled population is normally distributed, s = 100,  n = 25,  X ̅= 225, α  = 0.05,  Ha: µ > 220.  The critical value of the test statistic is ______________ .
a.    2.0639                    b.    1.7081
c.    1.7109                    d.     1.96

_____8. You perform a hypothesis test about a population mean on the basis of the following information: n = 50, X ̅= 100, α = 0.05, s = 30, Ha: µ < 110.  The computed value of the test statistic is _____________ .
a.    -2.3570                b.    -1.645
c.    2.3570                d.    4.24264

_____9. What is Z_0 score for P(Z≥Z_0) = 0.0708?
    1.47                    c.   1.80
    1.35                    d.   1.41

_____10. The random variable x has a normal distribution with μ = 40 and σ^2 = 36. What is the value of x if P(X≥X_0) = 0.40?
a.  47.86                    c. 49.85
b.  41.50                    d. 45.73

Part B: True or False (11-20)
_____11. A normal distribution is a distribution of discrete data that produces a bell-shaped.
_____12. The mean of the discrete probability distribution for a discrete random variable is called its expected value.
_____13. A random variable is a variable that can take different values according to the outcome of an experiment, and it can be either discrete or continuous.
_____14. The variance is the expected value of the squared difference between the random variable and its mean.
_____15. If the critical values of the test statistic z is ±1.96, they are the dividing points between the areas of rejection and non-rejection.
_____16. For the continuity correction, the normal distribution is continuous and the binomial is discrete.
_____17. The binomial probability table gives probability for value of p greater than 0.5.
_____18. The H_o cannot be written without having an equal sign.
_____19. For the normal distribution, the observations closer to the middle will occur with increasing frequency.
_____20. One assumption in testing a hypothesis about a proportion is that an outcome of an experiment can be classified into two mutual categories, namely, a success or a failure.

Part C: Answer the following questions (21-29)

    Explain the differences between discrete random variable and continuous random variable.

    What are the characteristics of discrete probability distribution?

    When should the z-test be used and when should t-test be used?

    What is the purpose of hypothesis testing?

    Can you prove the null? Why?

    What is Type I error?

    What is Type II error?
    Explain Sampling distribution of the mean

    Explain Central limit theorem

Part D: Fill in the blank (30-40)

    The purpose of hypothesis testing is to aid the manager or researcher in reaching a (an) __________________ concerning a (an) _______________ by examining the data contained in a (an) _______________ from that ____________________.

    A hypothesis may be defined simply as __________________________________________.

    There are two statistical hypotheses. They are the _________________ hypothesis and the _________________ hypothesis.

    The statement of what the investigator is trying to conclude is usually placed in the _________________ hypothesis.

    If the null hypothesis is not rejected, we conclude that the alternative _________________.

    If the null hypothesis is not rejected, we conclude that the null hypothesis _________________.

    The probability of committing a Type I error is designated by the symbol ____________, which is also called the ___________________.

    Values of the test statistic that separate the acceptance region from the rejection are called _________________ values.

    The following is a general statement of a decision rule: If, when the null hypothesis is true, the probability of obtaining a value of the test statistic as_______________ as or more _____________ than that actually obtained is less than or equal to , the null hypothesis is________________.  Otherwise, the null hypothesis is ______________________ .

 
    The probability of obtaining a value of the test statistic as extreme as or more extreme than that actually obtained, given that the tested null hypothesis is true, is called ____________ for the ________________test.

    When one is testing H0: µ= µ0 on the basis of data from a sample of size n from a normally distributed population with a known variance of σ2, the test statistic is ____________________________________________________.
Part E: Must show all your work step by step in order to receive the full credit; Excel is not allowed. (41-53)

    Ten trials are conducted in a Bernoulli process in which the probability of success in a given trail is 0.4. If x = the number of successes, determine the following.

 
a) E(x)      

    b) σ_x

c) P (x = 5)

    d) P (4 ≤ x ≤ 8)

e) P (x > 4)

    Work problem number 5 on page 6-14 (a-e).
a)         
    

    Work problem number 9 on page 6-28 (a-f).
a)

    b)
c)

    d)
e)

    f)

    Use problem number 4 on page 6-22 to fill in the table and answer the following questions (a-c).
X    P[X=x]    (X)(P[X=x])    [X-E(X)]    [X-E(X)]2    [X-E(X)]2 P[X=x]
0                    
1                    
2                    
3                    
4                    
5                    
6                    
Total                    

a) Expected value  
    

    b) Variance
c) Standard deviation

    Work problem number 5 on page 7-23 (a-f).(**Please draw the graph)

    Show your work    Please draw graph

    Work problem number 9 on page 7-47 (a-f). (** Please draw the graph)

    Find the following probabilities:(**Please draw the graph)
    Show your work    Please draw graph
a.    P(-1.4 < Z < 0.6)                                     

b.    P(Z > -1.44)                                            
    

c.    P(Z < 2.03)                                              

d.    P(Z > 1.67)

e.    P(Z < 2.84)

f.    P(1.14 < Z < 2.43)

    Find the Z scores for the following normal distribution problems.(** Please draw the graph)
    Show your work    Please draw graph
a.    µ = 604, σ = 56.8, P(X ≤ 635)

b.    µ = 48, σ2 = 144, P(X < 20)

c.    µ = 111, σ = 33.8, P(100 ≤ X ≤ 150)
    

d.    µ = 264, σ2 = 118.81, P(250 < X < 255)
    
e.    µ = 37, σ = 4.35, P(X > 35)
    
f.    µ = 156, σ = 11.4, P(X ≥ 170)
    

    Work problem on number 11 (a - f) on page 7-47 (a-f). (** Please draw the graph)
    Show your work    Please draw graph
a.        

Ho: µ ≥ 10
Ha: µ < 10
A sample of 50 provides a sample mean of 9.46 and sample variation of 4.
    Use Z or T test? And why?        At α = 0.05, what is the rejection rule?
    Compute the value of the test statistic.

        What is the p-value?
    What is your conclusion?

    Consider the following data drawn from a normal distribution population:

4    8    12    11    14    6    12    8    9    5
Construct 95% confidence interval using the above information and answer the following questions.

    What is sample mean        What is sample standard deviation
    Use Z or T test? And why?        At At 95% confidence interval, what is the rejection rule?

    Compute the value of the test statistic.

        What is α associated with this question?
    Interpret the confidence interval