The t test for two independent samples - One-tailed example using tables

Most engaged couples expect or at least hope that they will have high levels of marital satisfaction. However, because 54% of first marriages end in divorce, social scientists have begun investigating influences on marital satisfaction. [Data source: This data was obtained from the National Center for Health Statistics.]

Suppose a clinical psychologist sets out to look at the role of intactness of family of origin in relationship longevity. He decides to measure marital satisfaction in a group of couples from divorced families and a group of couples from intact families. He chooses the Marital Satisfaction Inventory, because it refers to “partner” and “relationship” rather than “spouse” and “marriage,” which makes it useful for research with both traditional and nontraditional couples. Higher scores on the Marital Satisfaction Inventory indicate greater satisfaction. There is one score per couple. Assume that these scores are normally distributed and that the variances of the scores are the same among couples from divorced families as among couples from intact families.

Q. The psychologist thinks that couples from divorced families will have less relationship satisfaction than couples from intact families. He identifies the null and alternative hypotheses as:

H₀: μcouples from divorced families couples from divorced families ( ) ?? μcouples from intact families couples from intact families

H₁: μcouples from divorced families couples from divorced families ( ) ?? μcouples from intact families couples from intact families

Q. This is a ( ) tailed test??

The Psychologist collects the data. A group of 30 couples from divorced families scored an average of 21.5 with a sample standard deviation of 10 on the Marital Satisfaction Inventory. A group of 27 couples from intact families scored an average of 25.8 with a sample standard deviation of 9. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom.

Q. The degrees of freedom are ( ) ?? .

Q. With α = 0.01, the critical t-score (the value for a t-score that separates the tail from the main body of the distribution, forming the critical region) is (-2.423 -2.668 2.326 2.054) ?? .

Q. To calculate the t statistic, you first need to calculate the estimated standard error of the difference in means. To calculate this estimated standard error, you first need to calculate the pooled variance. The pooled variance is (2.5308 89.3929 44.8423 91.0182 ) ?? Q. The estimated standard error of the difference in means is (2.5308 89.3929 91.0182 1.1934) ??   

Q. Calculate the t statistic. The t statistic is ( -1.50 -1.40 -1.70 -2.25) ??   

Q. The t statistic (does/does not) ??   lie in the critical region for a one-tailed hypothesis test. Therefore, the null hypothesis is (rejected/not rejected) ??

Q. The psychologist (can/can not)  ??   conclude that couples from divorced families have less relationship satisfaction than couples from intact families.

Ray Liu is a sales rep in northern California for high-end golf equipment. Each day, he makes 10 sales calls (n = 10). The chance of making a sale on each call is thought to be 25% (p = 0.25).

Assume this problem is appropriate for the binomial probability distribution.

A. E(x) =
Enter your answer rounded to 1 decimal in the format 1.2, with no other text or symbols.

B. σ_x =
Enter your answer rounded to 1 decimal in the format 1.2, with no other text or symbols.

C. Compute the following probabilities for Ray's calls tomorrow. Enter your responses in decimal format rounded to 3 decimals, in the format 0.123 with no other text or symbols.

The probability of making exactly 3 sales =

The probability of making fewer than 4 sales =

The probability of making more than 5 sales =

STEP 1: You are doing research on balance and fitness. To complete this research you will need a watch with a second hand. Identify a random sample of n = 12 men and n = 8 women. You must answer this question: How do you establish that this sample is truly random?

STEP 2: Have each subject perform the following task:

1. Have the subjects stand with their hands at their side, raise one knee, cross their ankle over the other knee, squat and bring their hands palms together in front of their chest. Time the subject until they put their foot back down on the floor.

b) Ask the following questions:

                        i) How many days per week do they exercise?

                        ii) What is their favorite exercise?

STEP 3: You will analyze your data and compute the following statistics for each group:

1) The Mean and standard deviation of the number of seconds the subject stayed balanced

2) The Median number of days per week exercised

3) The Mode of the favorite exercise

4) The 90% confidence interval of the mean

STEP 4: Construct a complete hypothesis test and determine if the two groups have significantly different balance using α = 0.05.

STEP 5: Write a one page introduction to your research, discuss how you selected your sample (is it a random sample?) and write a one page conclusion. Present your data in an organized manner.

Great white sharks are big and hungry. Here are the lengths in feet of 44 great whites.

18.6 16.3 13.3 19.0

12.5 16.5 15.7 16.4

18.5 17.8 14.1 23.0

16.6 16.1 16.8 16.6

15.6 12.7 9.3 13.7

18.4 18.0 18.3 13.4

14.4 13.7 13.3 15.8

15.7 12.1 13.7 19.5

14.8 15.3 15.4 18.6

17.6 14.8 15.9 13.2

12.1 12.5 13.4 16.9

(a) Examine these data for shape, center, spread and outliers. The distribution is reasonably Normal except for one outlier in each direction. Because these are not extreme and preserve the symmetry of the distribution, use of the tprocedures is safe with 44 observations. (Round your answer for x to two decimal places and your answer for s to three decimal places.)

x	=  ft
s	=  ft

(b) Give a 95% confidence interval for the mean length of great white sharks. (Round your answers to two decimal places.)
to  ft

Based on this interval, is there significant evidence at the 5% level to reject the claim "Great white sharks average 20 feet in length"?

Since 20 ft does not fall in this interval, we reject this claim.Since 20 ft does not fall in this interval, we accept this claim.    Since 20 ft does fall in this interval, we accept this claim.Since 20 ft does fall in this interval, we reject this claim.

(c) Before accepting the conclusions of (b), you need more information about the data. What would you like to know? (Select all that apply.)

Were they all male?Were these all full-grown sharks?Are these sharks dangerous?Were these all great white sharks?Had these sharks just eaten?Can these numbers be considered an SRS from this population?

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a) Suppose n = 27 and p = 0.26. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.)

np =

nq =

______ (Yes, or No), p̂ ______ (cannot, or can be) approximated by a normal random variable because ______ (np and nq do not exceed, np exceeds, np does not exceed, nq does not exceed, nq exceeds, or both np and nq exceed)

What are the values of μ_p̂ and σ_p̂? (Use 3 decimal places.)

μp̂ =

σp̂ =

(b) Suppose n = 25 and p = 0.15. Can we safely approximate p̂ by a normal distribution? Why or why not?
_____(Yes, or No), p̂  _______ (can, or cannot) be approximated by a normal random variable because _______ (np and nq do not exceed, np exceeds, np does not exceed, nq does not exceed, nq exceeds, or both np and nq exceed)
(c) Suppose n = 57 and p = 0.21. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.)

np =

nq =

_____(Yes, or No), p̂  _______ (can, or cannot) be approximated by a normal random variable because _______ (np and nq do not exceed, np exceeds, np does not exceed, nq does not exceed, nq exceeds, or both np and nq exceed)

What are the values of μ_p̂ and σ_p̂? (Use 3 decimal places.)

μp̂ =

σp̂ =

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. Method 1: Use the Student's t distribution with d.f. = n − 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution. This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution. Consider a random sample of size n = 41, with sample mean x = 44.9 and sample standard deviation s = 6.5.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit 43.19 42.85 42.16   

upper limit   46.61 46.95 47.65

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit 43.23 42.91 42.28

upper limit 46.57 46.89 47.52

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are shorter.

No. The respective intervals based on the t distribution are longer.

Yes. The respective intervals based on the t distribution are shorter.

Yes. The respective intervals based on the t distribution are longer. (This answer is correct)

(d) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%

lower limit

upper limit

(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

90% 95% 99%
lower limit

upper limit

1. In a poll, 360 students were tested about their interests in mathematics and statistics, the results are presented in the table below. Please determine at 0.01 level of significance whether a student’s interest in mathematics and independent of his/her interest in statistics.

Interest in statistics		Interest in mathematics
		Low	Average	High
	Low	63	42	15
	Average	58	61	31
	High	14	47	29

1. In this scenario which chi-square test does he need to perform?
2. Please construct the null and alternative hypotheses, and specify the type of test it is (e.g., two-sided, left or right sided test)
3. Please compute the expected counts for each cell of the contingency table
4. Please compute the test statistics.
5. Please determine the degree of freedom, and then compute the p-value (you are using p-value approach for this task)
6. Please state your conclusions.

The Customer Service Center in a large New York department store has determined that the amount of time spent with a customer about a complaint is normally distributed, with a mean of 8.5 minutes and a standard deviation of 2.4 minutes. What is the probability that for a randomly chosen customer with a complaint, the amount of time spent resolving the complaint will be as follows. (Round your answers to four decimal places.)

(a) less than 10 minutes


(b) longer than 5 minutes


(c) between 8 and 15 minutes

The following null and alternative hypotheses have been stated: H0: µ1 - µ2 = 0 HA: µ1 - µ2 = ø to test the null hypothesis, random samples have been selected from the two normally distributed populations with equal variances. the following sample data were observed. sample from population 1: 33, 29, 35, 39, 39, 41, 25, 33, 38. sample from population 2: 46, 43, 42, 46, 44, 47, 50, 43, 39. the test null hypothesis using an alpha level equal to 0.05.

Compute the following binomial probabilities using the table of Cumulative Binomial Probabilities. Give your answer to 3 places past the decimal.

c) Binomial pmf value: b(10; 15, 0.3)  

d) Binomial pmf value: b(11; 20, 0.4)

e) P(2 ≤ X ≤ 7) when X ~ Bin(15, 0.2)  

f) P(X ≥ 9) when X ~ Bin(15, 0.2)

g) P(6 < X ≤ 9) when X ~ Bin(15, 0.2)  

Question 7 Moe hired two separate trios of musicians to play at her tavern on Friday night. Each trio has one piano player, one guitar player, and one sax player. The guitar players have a 86% chance of showing up, the piano players have a 50% chance of showing up, and the saxophone players have a 46% chance of showing up. The two trios have completely different play lists, so individual musicians can't substitute for each other. What is the probability that at least one trio will play on Friday night? (Use four decimal places. Enter answer without a percent sign, e.g. 50% would be entered as .5)

Correct Answer 0.3565

Question 8 Moe hired two separate trios of musicians to play at her tavern on Friday night. Each trio has one piano player, one guitar player, and one sax player. The guitar players have a 93% chance of showing up, the piano players have a 65% chance of showing up, and the saxophone players have a 45% chance of showing up. The two trios know all the same songs so individual musicians can substitute for each other. What is the probability that at least one trio will play on Friday night? (Use four decimal places. Enter answer without a percent sign, e.g. 50% would be entered as .5) Correct Answer 0.6091

An investor has a certain amount of money available to invest now. Three alternative investments are available. The estimated profits ($) of each investment under each economic condition are indicated in the following payoff table:

INVESTMENT SELECTION

EVENT A B C

Economy declines 500 -2000 -7000

No change 1000 2000 -1000

Economy expands 2000 5000 20000

Based on his own past experience, the investor assigns the following probabilities to each economic condition:

P (Economy declines) = 0.30

P (No change) = 0.50

P (Economy expands) = 0.20

a. determine the optimal action based on the maximax criterion.

b. determine the optimal action based on the maximin criterion.

c.compute the the expected monetary value for each investment

d. compute the expected opportunity loss for each investment

e. based on (c) or (d) which investment would you choose, and why?

Thirty-five small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 40.7 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit
upper limit
margin of error

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

(a) As the confidence level increases, the margin of error decreases.

(b) As the confidence level increases, the margin of error remains the same.    

(c) As the confidence level increases, the margin of error increases.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

(a) As the confidence level increases, the confidence interval remains the same length.

(b) As the confidence level increases, the confidence interval decreases in length.    

(c) As the confidence level increases, the confidence interval increases in length.

In a random sample of​ males, it was found that

26

write with their left hands and

214

do not. In a random sample of​ females, it was found that

62

write with their left hands and

438

do not. Use a

0.05

significance level to test the claim that the rate of​ left-handedness among males is less than that among females. Complete parts​ (a) through​ (c) below.

a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of males and the second sample to be the sample of females. What are the null and alternative hypotheses for the hypothesis​ test?

A.

Upper H 0

​:

p 1

greater than or equalsp 2

Upper H 1

​:

p 1

not equalsp 2

B.

Upper H 0

​:

p 1

not equalsp 2

Upper H 1

​:

p 1

equalsp 2

C.

Upper H 0

​:

p 1

equalsp 2

Upper H 1

​:

p 1

not equalsp 2

D.

Upper H 0

​:

p 1

equalsp 2

Upper H 1

​:

p 1

greater thanp 2

E.

Upper H 0

​:

p 1

less than or equalsp 2

Upper H 1

​:

p 1

not equalsp 2

F.

Upper H 0

​:

p 1

equalsp 2

Upper H 1

​:

p 1

less thanp 2

Identify the test statistic.

zequals

nothing

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-valueequals

nothing

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

The​ P-value is

▼

less than

greater than

the significance level of

alpha

equals0.05

​,

so

▼

reject

fail to reject

the null hypothesis. There

▼

is sufficient

is not sufficient

evidence to support the claim that the rate of​ left-handedness among males is less than that among females.

b. Test the claim by constructing an appropriate confidence interval.

The

90

​%

confidence interval is

nothing

less thanleft parenthesis p 1 minus p 2 right parenthesisless thannothing

.

​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits

▼

do not include

include

​0, it appears that the two rates of​ left-handedness are

▼

not equal.

equal.

There

▼

is sufficient

is not sufficient

evidence to support the claim that the rate of​ left-handedness among males is less than that among females.

c. Based on the​ results, is the rate of​ left-handedness among males less than the rate of​ left-handedness among​ females?

A.

The rate of​ left-handedness among males

does

appear to be less than the rate of​ left-handedness among females because the results are statistically significant.

B.

The rate of​ left-handedness among males

does

appear to be less than the rate of​ left-handedness among females because the results are not statistically significant.

C.

The rate of​ left-handedness among males

does not

appear to be less than the rate of​ left-handedness among females.

D.

The results are inconclusive.

Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. The following is data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Assume σ ≈ 9.1 thousand dollars.

46.7	45.3	32.7	38.3	39.7	46.7	46.0	40.4	42.5	33.0	33.6
36.9	27.0	47.1	33.8	28.1	28.5	29.1	36.5	36.1	26.9	27.8
28.8	29.3	31.5	31.7	31.1	38.0	32.0	31.7	32.9	23.1	54.9
43.8	36.9	31.9	25.5	23.2	29.8	22.3	26.5	26.7

(a) Use a calculator or appropriate computer software to find x for the preceding data. (Round your answer to two decimal places.)
__________ thousand dollars

(b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a 75% confidence interval for μ, the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit	thousand dollars
upper limit	thousand dollars

(c) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    

No. This confidence interval suggests that the bank profits are less than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(d) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

No. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    

Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(e) Find a 90% confidence interval for μ, the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit	thousand dollars
upper limit	thousand dollars