Deck 10: T Tests, Two-Way Tables, and Anova

Question 1

A simple random sample from a normal distribution is taken in order to obtain a 95% confidence interval for the population mean. If the sample size is 11, the sample mean x is 32, and the sample standard deviation s is 12.6, what is the margin of error? Show your answer to 2 decimal places.

Answer 1

$$\mathrm { df } = 8 ; E = 2.228 \frac { 12.6 } { \sqrt { 11 } } = 8.46$$

Question 2

-The t distribution can be used when finding a confidence interval for the population mean whenever the sample size is small.

Answer 2

False

Question 3

A simple random sample from a normal distribution is taken in order to obtain a 95% confidence interval for the population mean. If the sample size is 8, the sample mean x is 22, and the sample standard deviation s is 6.3, what is the margin of error? Show your answer to 2 decimal places.

Answer 3

$$\mathrm { df } = 7 ; E = 2.365 \frac { 6.3 } { \sqrt { 8 } } = 5.869$$

Question 4

The partial t-table below can be used where necessary.


-The t distribution can be used when finding a confidence interval for the mean with a small sample whenever the data comprise a simple random sample.

Question 5

-Data from the test in Problem 5 resulted in a sample mean of 163.2 yards with a sample standard deviation of 5.8 yards. Assuming normality, carry out a hypothesis test at the 0.05

Question 6

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

-Find the value of the ?2 statistic for the data .

Question 7

One hundred people are selected at random and tested for colorblindness to determine whether gender and colorblindness are independent. The following counts were observed.

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$
If gender and colorblindness are independent, find the expected values corresponding to the four combinations of gender and colorblindness, and enter them in the table below along with row and column totals.

$\begin{array}{|l|l|l|l|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & & & \\\hline \text { Female } & & & \\\hline \text { Total } & & & \\\hline\end{array}$

Question 8

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

-State the null and alternative hypothesis for the test associated with the data

Question 9

A 95% confidence interval for the mean of a normal population is found to be 15.6 < µ < 25.2. What is the margin of error?

Question 10

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

-The critical value of ?² for a 2 x 2 table using a 0.05 significance level is 3.841. If the value of the ?² statistic iS 3.427, state your conclusion about the relationship between gender and colorblindness.

Question 11

A 95% confidence interval for the mean of a normal population is found to be 13.2 < µ < 22.4. What is the margin of error?

Question 12

The analysis of the data in Problem 9 resulted in the following output from Excel.
Anova: Single Factor

$\text { SUMMARY }$
$\begin{array}{lrrrr}\hline\text { Groups }&\text { Count } &{\text { Sum }} & {\text { Average }} &{\text { Variance }} \\\hline A&4 & 119 & 29.75 & 12.25 \\B&4 & 100 & 25 & 13.33333 \\C&4 & 94 & 23.5 & 27 \\\hline\end{array}$

$\begin{array}{lrrrrrr}\text { ANOVA }\\\hline \text { Source of Variation } & \text { SS } & d f & M S & F & \text { P-value } & \text { F crit } \\\hline\text { Between Groups } & 85.16667 & 2 & 42.58333 & 2.429477 & 0.143319 & 4.256492 \\\text { Within Groups } & 157.75 & 9 & 17.52778 & & &\\\\\text { Total}& 242.9167&11\\\end{array}$

If the significance level for the test is 0.05, which conclusion below is correct?

A) The data do not provide sufficient evidence to conclude that the groups A, B, and C are somehow related.
B) The data do not provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
C) The data do not provide sufficient evidence to conclude that the population variances of groups A, B, and C are different.
D) The data do provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
E) None of the above.

Question 13

$$\begin{array} { | l | c | c | c | } \hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 7 & 53 & 60 \\\hline \text { Female } & 1 & 39 & 40 \\\hline \text { Total } & 8 & 92 & 100 \\\hline\end{array}$$

-Find the value of the ?2 statistic for the data

Question 14

The following data were analyzed using one-way analysis of variance.
$$\begin{array} { c c c } \mathrm { A } & \mathrm { B } & \mathrm { C } \\\hline 34 & 27 & 19 \\26 & 23 & 31 \\31 & 29 & 22 \\28 & 21 & 22\end{array}$$ ` Which one of the following statements is correct?

A) The purpose of the analysis is to determine whether the groups A, B, and C are independent.
B) The purpose of the analysis is to test the hypothesis that the population means of the three groups are equal.
C) The purpose of the analysis is to test the hypothesis that the population variances of the three groups are equal.
D) The purpose of the analysis is to test the hypothesis that the sample means of the three groups are equal.

Question 15

A golfer wished to find a ball that would travel more than 160 yards when hit with his 7-iron with a club speed of 90 miles per hour. He had a golf equipment lab test a low compression ball by having a robot swing his club 8 times at the required speed. State the null and alternative hypotheses for this test.

Question 16

One hundred people are selected at random and tested for colorblindness to determine whether gender and colorblindness are independent. The following counts were observed.
$$\begin{array} { | l | c | c | c | } \hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 7 & 53 & 60 \\\hline \text { Female } & 1 & 39 & 40 \\\hline \text { Total } & 8 & 92 & 100 \\\hline\end{array}$$
If gender and colorblindness are independent, find the expected values corresponding to the four combinations of gender and colorblindness, and enter them in the table below along with row and column totals.
$$\begin{array} { | l | l | l | l | } \hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & & & \\\hline \text { Female } & & & \\\hline \text { Total } & & & \\\hline\end{array}$$

Question 17

-Resulted in a sample mean of 171.6 yards with a sample standard deviation of 2.4 yards. Assuming normality, carry out a hypothesis test at the 0.05 significance

Question 18

The following data were analyzed using one-way analysis of variance. $$\begin{array} { c c c } \text { A } & \text { B } & \text { C } \\\hline 24 & 27 & 19 \\26 & 23 & 31 \\31 & 29 & 22 \\28 & 21 & 22\end{array}$$ ` Which one of the following statements is correct?

A) The purpose of the analysis is to determine whether the groups A, B, and C are independent.
B) The purpose of the analysis is to test the hypothesis that the population means of the three groups are equal.
C) The purpose of the analysis is to test the hypothesis that the population variances of the three groups are equal.
D) The purpose of the analysis is to test the hypothesis that the sample means of the three groups are equal.

Question 19

A golfer wished to find a ball that would travel more than 170 yards when hit with his 6-iron with a club head speed of 90 miles per hour. He had a golf equipment lab test a low compression ball by having a robot swing his club 12 times at the required speed. State the null and alternative hypotheses for this test.

Question 20

$$\begin{array} { | l | c | c | c | } \hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 7 & 53 & 60 \\\hline \text { Female } & 1 & 39 & 40 \\\hline \text { Total } & 8 & 92 & 100 \\\hline\end{array}$$

-The critical value of ? ² for a 2 x 2 table using a 0.05 significance level is 3.841. If the value of the ? ² statistic is 3.179, state your conclusion about the relationship between gender and colorblindness.

Question 21

The critical value of χ2 for a 2 x 2 table using a 0.05 significance level is 3.841. If the value of the
χ2 statistic in Problem 8 had been 4.216, state your conclusion about the relationship between
gender and colorblindness.

Question 22

The margin of error in estimating the population mean of a normal population is E = 6.3 when the sample standard deviation is 8.2. If the sample standard deviation were increased to 9.1 and the sample size stayed the same, would the margin of error be larger or smaller than 6.3? Explain your answer.

Question 23

The following data were analyzed using one-way analysis of variance. $$\begin{array} { c c c } \text { A } & \text { B } & \text { C } \\\hline 34 & 27 & 19 \\26 & 23 & 21 \\31 & 29 & 22 \\28 & 21 & 12\end{array}$$ ` Which one of the following statements is correct?

A) The purpose of the analysis is to determine whether the groups A, B, and C are independent.
B) The purpose of the analysis is to test the hypothesis that the population means of the three groups are equal.
C) The purpose of the analysis is to test the hypothesis that the population variances of the three groups are equal.
D) The purpose of the analysis is to test the hypothesis that the sample means of the three groups are equal.

Question 24

-Resulted in a sample mean of 184.2 yards and a sample standard deviation of 5.8 yards. Assuming normality, carry out a hypothesis test at the 0.05 significance

Question 25

A simple random sample from a normal distribution is taken in order to obtain a 95% confidence interval for the population mean. If the sample size is 12, the sample mean x
is 22, and the sample standard deviation s is 5.5, what is the margin of error? Show your answer to 2 decimal places.

Question 26

The following data were analyzed using one-way analysis of variance. $$\begin{array} { c c c } \text { A } & \text { B } & \text { C } \\\hline 29 & 27 & 19 \\26 & 23 & 21 \\25 & 29 & 25 \\28 & 21 & 17\end{array}$$ ` Which one of the following statements is correct?

A) The purpose of the analysis is to determine whether the groups A, B, and C are independent.
B) The purpose of the analysis is to test the hypothesis that the population means of the three groups are equal.
C) The purpose of the analysis is to test the hypothesis that the population variances of the three groups are equal.
D) The purpose of the analysis is to test the hypothesis that the sample means of the three groups are equal.

Question 27

A 95% confidence interval for the mean of a normal population is found to be 15.6 < µ < 24.8. What is the margin of error?

Question 28

The analysis of the data in Problem 9 resulted in the following output from Excel.

Anova: Single Factor

$\text { SUMMARY }$
$\begin{array}{lrrrr}\hline\text { Groups }&\text { Count } &{\text { Sum }} & {\text { Average }} &{\text { Variance }} \\\hline A&4 & 108 &27 & 3.333333 \\B&4 & 100 & 25 & 13.33333 \\C&4 &82 & 20.5 & 11.66667 \\\hline\end{array}$

$\begin{array}{lrrrrrr}\text { ANOVA }\\\hline \text { Source of Variation } & \text { SS } & d f & M S & F & \text { P-value } & \text { F crit } \\\hline\text { Between Groups } & 88.66667& 2 & 44.33333&4.694118 &0.040148 & 4.256492\\\text { Within Groups } & 85& 9 & 9.444444 & & &\\\\\text { Total}& 173.6667&11\\\hline\end{array}$

If the significance level for the test is 0.05, which conclusion below is correct?

A) The data provide sufficient evidence to conclude that the groups A, B, and C are somehow related.
B) The data provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
C) The data provide sufficient evidence to conclude that the population variances of groups A, B, and C are different.
D) The data do not provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
E) None of the above.

Question 29

A 95% confidence interval for the mean of a normal population is found to be 17.6 < µ < 23.6.
What is the margin of error?

Question 30

A simple random sample from a normal distribution is taken in order to obtain a 95% confidence
interval for the population mean. If the sample size is 12, the sample mean x
is 32, and the
sample standard deviation s is 7.5, what is the margin of error? Show your answer to 2 decimal
places.

Question 31

State the null and alternative hypothesis for the test associated with the data in Problem 7.

Question 32

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

-The critical value of ?² for a 2 x 2 table using a 0.05 significance level is 3.841. If the value of the ?² statistic is 4.613, state your conclusion about the relationship between gender and colorblindness.

Question 33

$\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

-State the null and alternative hypothesis for the test associated with the data.

Question 34

One hundred people are selected at random and tested for colorblindness to determine whether
gender and colorblindness are independent. The following counts were observed.
Colorblind Not Colorblind Total $\begin{array}{|l|c|c|c|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & 8 & 52 & 60 \\\hline \text { Female } & 2 & 38 & 40 \\\hline \text { Total } & 10 & 90 & 100 \\\hline\end{array}$

If gender and colorblindness are independent, find the expected values corresponding to the four combinations of gender and colorblindness, and enter them in the following table along with row

$\begin{array}{|l|l|l|l|}\hline & \text { Colorblind } & \text { Not Colorblind } & \text { Total } \\\hline \text { Male } & & & \\\hline \text { Female } & & & \\\hline \text { Total } & & & \\\hline\end{array}$

Question 35

The t distribution can be used when finding a confidence interval for the population mean
with a small sample whenever the sample comes from a symmetric population.

Question 36

level and determine whether the ball meets the golfer's requirements. Use the partial t-table
above.
One hundred people are selected at random and tested for colorblindness to determine whether
gender and colorblindness are independent. The following counts were observed.

Question 37

Find the value of the χ2 statistic for the data in Problem 7.
131

Question 38

-The t distribution can be used when finding a confidence interval for the population mean with a small sample anytime the population standard deviation is unknown.

Question 39

A golfer wished to find a ball that would travel more than 180 yards when hit with his 5-iron witha club speed of 90 miles per hour. He had a golf equipment lab test a low compression ball by having a robot swing his club 7 times at the required speed. State the null and alternative hypotheses for this test.

Question 40

The analysis of the data in Problem 9 resulted in the following output from Excel.

Anova: Single Factor

$\text { SUMMARY }$
$\begin{array}{lrrrr}\hline\text { Groups }&\text { Count } &{\text { Sum }} & {\text { Average }} &{\text { Variance }} \\\hline A&4 & 109 &27.25 & 8.916667 \\B&4 & 100 & 25 & 13.33333 \\C&4 & 94 & 23.5 & 27 \\\hline\end{array}$

$\begin{array}{lrrrrrr}\text { ANOVA }\\\hline \text { Source of Variation } & \text { SS } & d f & M S & F & \text { P-value } & \text { F crit } \\\hline\text { Between Groups } & 28.5 & 2 & 14.25 &0.86802 &0.452161 & 4.256492 \\\text { Within Groups } & 147.75 & 9 & 16.41667 & & &\\\\\text { Total}& 176.25&11\\\hline\end{array}$

If the significance level for the test is 0.05, which conclusion below is correct?

A) The data do not provide sufficient evidence to conclude that the groups A, B, and C are somehow related.
B) The data do not provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
C) The data do not provide sufficient evidence to conclude that the population variances of groups A, B, and C are different.
D) The data do provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
E) None of the above.

Question 41

The analysis of the data in Problem 9 resulted in the following output from Excel. Anova: Single Factor If the significance level for the test is 0.05, which conclusion below is correct?

A) The data do not provide sufficient evidence to conclude that the groups A, B, and C are somehow related.
B) The data do not provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
C) The data do not provide sufficient evidence to conclude that the population variances of groups A, B, and C are different.
D) The data do provide sufficient evidence to conclude that the population means of groups A, B, and C are different.
E) None of the above.