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Chapter 8: Q89SE (page 503)

List the assumptions necessary for each of the following inferential techniques:

a. Large-sample inferences about the difference\({\mu _1} - {\mu _2}\) between population means using a two sample z-statistic

Short Answer

Expert verified

a)

  • The two samples are independent.
  • The populations have any distribution.
  • The two samples are large sample. i.e. \({n_1} \ge 30\,and\,{n_2} \ge 30\)

Step by step solution

01

Given Information

Let \({x_1}\,and\,{x_2}\) are two populations.

\({\mu _1}\)is the mean for population 1.

\({\mu _2}\)is the mean for population 2.

\({\mu _1} - {\mu _2}\) is the difference between them.

02

Test statistic and confidence interval of large sample

The test statistic is given by

\(z = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - {D_0}}}{{\sqrt {\frac{{\sigma _1^2}}{{{n_1}}} + \frac{{\sigma _2^2}}{{{n_2}}}} }}\)

And the confidence interval is given by

\(\left( {{{\bar x}_1} - {{\bar x}_2}} \right) \pm {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\sigma _1^2}}{{{n_1}}} + \frac{{\sigma _2^2}}{{{n_2}}}} \)

03

Assumptions

  • The two samples are independent.
  • The populations have any distribution.
  • The two samples are large sample. i.e. \({n_1} \ge 30\,and\,{n_2} \ge 30\)

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Most popular questions from this chapter

Independent random samples from two populations with standard deviations 蟽1=2补苍诲蟽2=8, respectively, are selected. The sample sizes and the sample means are recorded in the following table:

Sample 1

Sample 2

n1=58x炉1=17.5

n2=62x炉2=16.23

a. Calculate the standard error of the sampling distribution for Sample 1.

b. Calculate the standard error of the sampling distribution for Sample 2.

c. Suppose you were to calculate the difference between the sample means (x1鈭�x2). Find the mean and standard error of the sampling distribution (x1鈭�x2).

d. Will the statistic (x1鈭�x2) be normally distributed?

Consider the discrete probability distribution shown here.

x

10

12

18

20

p

.2

.3

.1

.4

a. Calculate渭,蟽2 and蟽 .

b. What isP(x<15) ?

c. Calculate 渭卤2蟽 .

d. What is the probability that xis in the interval 渭卤2蟽 ?

Question: Deferred tax allowance study. A study was conducted to identify accounting choice variables that influence a manager鈥檚 decision to change the level of the deferred tax asset allowance at the firm (The Engineering Economist, January/February 2004). Data were collected for a sample of 329 firms that reported deferred tax assets in 2000. The dependent variable of interest (DTVA) is measured as the change in the deferred tax asset valuation allowance divided by the deferred tax asset. The independent variables used as predictors of DTVA are listed as follows:

LEVERAGE: x1= ratio of debt book value to shareholder鈥檚 equity

BONUS: x2 = 1 if firm maintains a management bonus plan,

0 if not

MVALUE: x3 = market value of common stock

BBATH: x4 = 1 if operating earnings negative and lower than last year,

0 if not

EARN: x5 = change in operating earnings divided by total assets

A first-order model was fit to the data with the following results (p-values in parentheses):

Ra2 = .280

y^=0.044+0.006x1-0.035x2-0.001x3+0.296x4+0.010x5

(.070) (.228) (.157) (.678) (.001) (.869)

  1. Interpret the estimate of the 尾 coefficient for x4.
  2. The 鈥淏ig Bath鈥� theory proposed by the researchers鈥� states that the mean DTVA for firms with negative earnings and earnings lower than last year will exceed the mean DTVA of other firms. Is there evidence to support this theory? Test using 伪 = .05.
  3. Interpret the value of Ra2.

Question: Independent random samples from approximately normal populations produced the results shown below.

Sample 1

Sample 2

52 33 42 4441 50 44 5145 38 37 4044 50 43

52 43 47 5662 53 61 5056 52 53 6050 48 60 55

a. Do the data provide sufficient evidence to conclude that (渭1-渭2)>10? Test using伪=0.1.

b. Construct a confidence interval for (渭1-渭2). Interpret your result.

Solar energy generation along highways. Refer to the International Journal of Energy and Environmental Engineering (December 2013) study of solar energy generation along highways, Exercise 8.39 (p. 481). Recall that the researchers compared the mean monthly amount of solar energy generated by east-west鈥� and north-south鈥� oriented solar panels using a matched-pairs experiment. However, a small sample of only five months was used for the analysis. How many more months would need to be selected to estimate the difference in means to within 25 kilowatt-hours with a 90% confidence interval? Use the information provided in the SOLAR file to find an estimate of the standard error required to carry out the calculation
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