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Chapter 8: Q4E (page 465)

To use the t-statistic to test for a difference between the means of two populations, what assumptions must be made about the two populations? About the two samples?

Short Answer

Expert verified

The T Statistic is used in a T-test to determine whether to accept as well as deny the empty hypothesis.

Step by step solution

01

Step-by-Step Solution Step 1: Definition of t-test.

The t-test is a statistical test used to compare the means of two groups. It helps us to determine whether there is a significant difference between the mean of the two groups or not.

The formula of the t-test is:

t=x炉鈭�渭蟽n

02

Assumptions to be made about populations and samples.

The common assumptions made when doing a t-test include those regarding the scale of measurement, random sampling, normality of data distribution, adequacy of sample size, and equality of variance in standard deviation.

  1. The observations in one sample are independent of the observations in the other sample.
  2. Both samples are approximately normally distributed.
  3. Both samples have approximately the same variance.
  4. Both samples were obtained using a random sampling method.

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Most popular questions from this chapter

The data for a random sample of 10 paired observations is shown below.

PairSample from Population 1

(Observation 1)

Sample from Population 2 (Observation 2)
12345678910
19253152493459471751
24273653553466512055

a. If you wish to test whether these data are sufficient to indicate that the mean for population 2 is larger than that for population 1, what are the appropriate null and alternative hypotheses? Define any symbols you use.

b. Conduct the test, part a, using伪=.10.

c. Find a 90%confidence interval for 渭d. Interpret this result.

d. What assumptions are necessary to ensure the validity of this analysis?

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?

The 鈥渨inner鈥檚 curse鈥� in transaction bidding. In transaction bidding, the 鈥渨inner鈥檚 curse鈥� is the miracle of the winning (or loftiest) shot price being above the anticipated value of the item being auctioned. The Review of Economics and Statistics (Aug. 2001) published a study on whether shot experience impacts the liability of the winner鈥檚 curse being. Two groups of a stab in a sealed-shot transaction were compared (1)super-experienced stab and (2) less educated stab. In the super-experienced group, 29 of 189 winning flings were above the item鈥檚 anticipated value; 32 of 149 winning flings were above the item鈥檚 anticipated value in the less-educated group.

  1. Find an estimate of p1, the true proportion of super educated stab who fell prey to the winner鈥檚 curse
  2. Find an estimate of p2, the true proportion of less-educated stab who fell prey to the winner鈥檚 curse.
  3. Construct a 90 confidence interval for p1-p2.
  4. d. Give a practical interpretation of the confidence interval, part c. Make a statement about whether shot experience impacts the liability of the winner鈥檚 curse being.

The data for a random sample of six paired observations are shown in the next table.

a. Calculate the difference between each pair of observations by subtracting observation two from observation 1. Use the differences to calculate d炉andsd2.

b. If 渭1补苍诲渭2are the means of populations 1 and 2, respectively, expressed 渭din terms of 渭1补苍诲渭2.

PairSample from Population 1

(Observation 1)

Sample from Population 2(Observation 2)
123456739648417247

c. Form a 95% confidence interval for 渭d.

d. Test the null hypothesis H0:渭d=0against the alternative hypothesis Ha:渭d鈮�0. Use伪=.05 .

Patron amenability to supply biomass. Relate to the Biomass and Energy (Vol. 36, 2012) study of the amenability of directors to supply biomass products similar to fat hay, Exercise8.20 (p. 469). Recall that independent samples of Missouri directors and Illinois directors were surveyed. Another aspect of the study concentrated on the service directors who were willing to supply. One essential service involves windrowing (mowing and piling) hay. Of the 558 Missouri directors surveyed, 187 were willing to offer windrowing. Of the 940 Illinois directors surveyed, 380 were willing to offer windrowing services. The experimenters want to know if the proportion of directors willing to offer windrowing services to the biomass request differs for the two areas, Missouri and Illinois.

a. Specify the parameter of interest to the experimenters.

b. Set up the null and indispensable suppositions for testing whether the proportion of directors willing to offer windrowing services differs in Missouri and Illinois.

c. A Minitab analysis of the data is given below. Detect the test statistic on the printout.

d. provide the rejection region for the test using a = .01.

e. Detect the p- the value of the test on the printout.

f. Make the applicable conclusion using both the p-value and rejection region approach. Your conclusions should agree.
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