(Solved by Humans)-Hi! If you could help me out with number 4 in the short answers

Question

Hi! If you could help me out with number 4 in the short answers section and number 2 in the long answer?section I would greatly appreciate it! Could you also show your work and the steps and formulas used so I can use them to study? Thank you!!

Econ 2810

Intro to Statistics and Computing In Economics

Spring 2016

Final Exam Review
Final Exam Time and Date: 10:20-12:20pm, May 10, 2016

True or False (15 points)
1. The standard error is equal to

?s .
n

2. As the number of degrees of freedom increase, the t distribution gets closer and closer to the normal
distribution.
3. The 95% confidence interval for the mean ? of a normal population with known standard deviation ?
is
(
)
?
X ? z0.05
.
n?1
4. For a two sided test, if | t |> t0.05 , then we can reject the null hypothesis at the 10% level of significance.
5. To use the t test statistic for statistical inference, the population variance (? 2 ) is assumed to be known.

Multiple Choice Questions (15 points)
1. The one-sample t statistic from a sample of n = 21 observations for the two-sided test of
H0 : ? = 60
H1 : ? ?= 60
has the value t = ?1.98. Based on this information:
a. we would reject the null hypothesis at ? = 0.10.
b. 0.025 < P ? value < 0.05
c. we would reject the null hypothesis at ? = 0.05
d. none of the above
2. The costs of four new cellular phones are $160, $215, $99 and $195. The standard error of the mean
of these four cellular phones is:
a. 50.86
b. 167.25
c. 12.71
d. 25.43
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3. We test the null hypothesis H0 : ? = 10 and the alternative H1 : ? ?= 10 for a normal population with
? = 4. A random sample of 16 observations is drawn from the population and we find the sample
mean of these observations is X = 12. The P ? value is approximately
a. 0.0228
b. 0.0456
c. 0.4886
d. 0.9772
4. If the degrees of freedom = 10, then t0.001 is
a. 3.169
b. 4.297
c. 2.764
d. 4.144
5. The pooled two sample t statistic can be used if
a. The two populations have unequal variances and equal sample sizes
b. The two populations have unequal variances and unequal sample sizes
c. The two populations have equal variances and equal sample sizes
d. None of the above are correct

Short Answer (40 points)
1. A researcher is studying the failure rate of restaurants. She selects a random sample of 200 restaurants
in large cities that opened within the last year. Following up on these restaurants, she finds that 80
had failed within five years. Calculate a 99% confidence interval for the proportion of restaurants that
fail within five years.
2. Based on a sample of the salaries of professors at a major university, you have performed a multiple regression relating salary to years of service and gender. The estimated multiple linear regression model is:
Salary = $45,000 + $3000(Years) + $4000(Gender) + $1000[(Years)(Gender)]
where Gender = 1 if the professor is male and Gender = 0 if the professor is female.
a. Using the above multiple linear regression equation, calculate the average salary of male professors
with three years of experience.
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3. Consider the following two way ANOVA table. Find the values for a, b, c, d, e, f.
Source
Brand

df
3

SS
3410

MS
1136.8

F
d

P ?value
0.000

Club

1

a

46443.9

e

0.000

Brand*Club

3

105.2

b

f

0.260

Error

24

590.4

24.6

Total

31

c

4. Independent random samples are selected from two normal populations. Information for the 2 samples
are in the table below:

n
X
S

Population 1
14
7.1
2.3

Population 2
11
8.4
2.9

If you wanted to test the means are, in other words you are testing the following hypotheses:
H0 : ?21 ? ?22 = 0
H1 : ?21 ? ?22 ?= 0
a. Setting ? = 0.05, do you have enough statistical evidence to reject the null hypothesis? Why or
why not?
b. Now you want to test whether the population variances are equal. Setting ? = 0.05, do you have
enough statistical evidence to reject that the population variances are equal?

Long Answer (30 points)
1. In a completely randomized design experiment, 10 individuals were randomly chosen for each of three
treatment groups and a quantity was measured for each unit within each group. In the first steps of
testing whether the means of the three groups are the same, the sum of squares for treatments was
calculated to be 3110 and the sum of squares for error was calculated to be 27000
a. Complete the following ANOVA table
Source
Treatment

df

SS

MS

Error
Total
b. Suppose you want to test whether the means are identical. Write down the null and alternative
hypotheses
c. Setting ? = 0.05, can you reject the null hypothesis you constructed in part b)? Why or why not?
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2. A random sample of 79 companies from the Forbes 500 list (which actually consists of nearly 800
companies) was selected, and the relationship between sales (in hundreds of thousands of dollars) and
profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear
regression model was used
Profitsi = ?0 + ?1 (Sales)i + ?i
where the deviations ?i were assumed to be independent and normally distributed. This model was
fit to the data using the method of least squares. The following results were obtained from statistical
software:
R2 = 0.662
s = 466.2
Variable
Constant
Sales

Parameter Est.
176.644
0.092498

Std. Err. of Parameter Est.
61.16
0.0075

a. What is the value of the intercept?
b. Construct a 90% confidence interval for the slope ?1
c. Suppose the researchers test the hypotheses:
H 0 : ?1 = 0
H 1 : ?1 > 0
Can you reject the null hypothesis if ? = 0.01?
d. Suppose we wish to predict the mean profits (in hundreds of thousands of dollars) for all companies
that had sales (in hundreds of thousands of dollars) of 500. Given that sales is 500, what is the
predicted value for mean profits?