Question 1

In a metal fabrication process, metal rods are produced to a specified target length of
15 feet. Suppose that the lengths are normally distributed. A quality control specialist
collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and
a standard deviation of 0.65 feet.
A. Describe the sampling distribution for the sample mean.
b. What is the standard error?
c. For 95% confidence, what is the margin of error?
d. Based on the sample results, create the 95% confidence interval and interpret.

Accepted Answer

a. Describe the sampling distribution for the sample mean.
The sampling distribution for the sample mean can be modeled using the $t$-distribution with 15 degrees of freedom.
b. What is the standard error?
$$S E ( \bar { y } ) = \frac { s } { \sqrt { n } } = \frac { 0.65 } { \sqrt { 16 } } = 0.1625 \mathrm { ft }$$
c. For $95 \%$ confidence, what is the margin of error?
$$t _ { n - 1 } ^ { * } \times S E ( \bar { y } ) = 2.131 \times 0.1625 f t = 0.346 f t$$
d. Based on the sample results, create the $95 \%$ confidence interval and interpret.
$$\bar { y } \pm t _ { n - 1 } ^ { * } \times S E ( \bar { y } ) = 14.8 \pm 0.346$$
The $95 \%$ confidence interval for true mean length is from $14.454$ to $15.146 \mathrm { ft }$. We are $95 \%$ confident that the average length of metal rods from this process is between $14.454$ and $15.146 \mathrm { ft }$.

Question 2

A sample of students from an introductory business course were polled regarding the
number of hours they spent studying for the last exam. All students anonymously
submitted the number of hours on a 3 by 5 card. There were 24 individuals in the one
section of the course polled. The data was used to make inferences regarding the other
students taking the course. The data are shown below: $$\begin{array} { | c | c | c | c | c | c | c | c | c | c | c | c | } 
\hline 4.5 & 22 & 7 & 14.5 & 9 & 9 & 3.5 & 8 & 11 & 7.5 & 18 & 20 \
\hline 7.5 & 9 & 10.5 & 15 & 19 & 2.5 & 5 & 9 & 8.5 & 14 & 20 & 8 \
\hline
\end{array}$$ 
A. Based on the sample results, find the 95% confidence interval.
b. Interpret the results.
c. Do you expect a 90% confidence interval to be wider or narrower and why?
d. A previous study of a large cross-section of students in this course showed that
students studies 12 hours per week. Are the results of this current study statistically
different than the assumption of 12 hours per week studying? Define the hypotheses and
compare the t-value to the critical t value, evaluating at α = 0.05.

Accepted Answer

a. Based on the sample results, find the $95 \%$ confidence interval.
The sample has a mean of $10.92$ hours and a standard deviation of $5.60$ hours.
$\bar { y } \pm t _ { n - 1 } ^ { * } \times S E ( \bar { y } ) = 10.92 \pm 2.069 \frac { 5.60 } { \sqrt { 24 } } = 10.92 \pm 2.365$
The $95 \%$ confidence interval for true mean age is $8.5$ to $13.3$ hours.
b. Interpret the results.
We are $95 \%$ confident that the average age of clients with recently paid policies is between $76.87$ and $80.33$ years.
c. Do you expect a $90 \%$ confidence interval to be wider or narrower and why?
A $90 \%$ confidence interval will be narrower because the margin of error will be smaller for a lower confidence level.
d. A previous study of a large cross-section of students in this course showed that students studies 12 hours per week. Are the results of this current study statistically different than the assumption of 12 hours per week studying? Define the hypotheses and compare the $t$-value to the critical $t$ value, evaluating at $\alpha = 0.05$.
Hypotheses: $\mathrm { H } _ { 0 } : \mu = 12 \mathrm { hrs } ; \mathrm { H } _ { \mathrm { A } } : \mu \neq 12 \mathrm { hrs }$
t-test: $t = \frac { 10.92 - 12 } { 5.6 / \sqrt { 24 } } = - 0.945$; the critical $\mathrm { t }$ value for $\mathrm { df } = 23$ at $0.05 = \pm 2.0686$
The calculated value is less than the critical value so the result is not statistically significant at $\alpha = 0.05$. There is no evidence that the level of studying has changed.

Question 3

A pharmaceutical company wants to answer the question whether it takes longer than 45 seconds for a drug in pill form to dissolve in the gastric juices of the stomach. A
Sample was taken from 8 patients taking the given drug in pill form and times for the pills
To be dissolved were measured. The mean was 45.212 sec for the sample data with a
Standard error of 0.580. Determine the P-value for this test.

Accepted Answer

A)   0.181
B)   0.724
C)   0.649
D)   0.362
E)   0.366 D

Question 4

Grandma Gertrude's Chocolates, a family owned business, has an opportunity to
supply its product for distribution through a large coffee house chain. However, the
coffee house chain has certain specifications regarding cacao content as it wishes to
advertise the health benefits (antioxidants) of the chocolate products it sells. In order to
determine the mean % cacao in its dark chocolate products, quality inspectors sample 36
pieces. They find a sample mean of 55% with a standard deviation of 4%.
A. Describe the sampling distribution for the sample mean.
b. What is the standard error?
c. What is the margin of error for 90% confidence?
d. What is the margin of error for 95% confidence?
e. Based on the sample results, find the 90% confidence interval and interpret.
f. Based on the sample results, find the 95% confidence interval and interpret.
g. For a more accurate determination of the mean weight, the quality control inspectors
wish to estimate it within 1% with 95% confidence. How many pieces of dark chocolate
should they sample?

Accepted Answer

The answer of Grandma Gertrude's Chocolates, a family owned business,...

Question 5

A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the
Ravioli is normally distributed. To make sure that the automatic filling process is on
Target, quality control inspectors take a sample of 25 ravioli and measure the weight of
Cheese filling. They find a sample mean weight of 15 grams with a standard deviation of
1)5 grams. What is the margin of error at 90% confidence?

Accepted Answer

A)   0.3 grams
B)   0.5133 grams
C)   0.8391 grams
D)   1.5 grams
E)   0.06 grams

Question 6

A large software development firm recently relocated its facilities. Top management
is interested in fostering good relations with their new local community and has
encouraged their professional employees to engage in local service activities. They wish
to determine the average number of hours the firm's professionals volunteer per month.
A random sample of 24 professionals reported the following number of hours: $$\begin{array} { | l | l | l | l | l | l | l | l | l | l | l | l | } 
\hline 12 & 13 & 14 & 14 & 15 & 15 & 15 & 16 & 16 & 16 & 16 & 16 \
\hline 17 & 17 & 17 & 18 & 18 & 18 & 18 & 19 & 19 & 19 & 20 & 21 \
\hline
\end{array}$$ 
A. Based on the sample results, find the 95% confidence interval and interpret.
b. For a more accurate determination, top management wants to estimate the average
number of hours volunteered per month by their professional staff to within one hour with
99% confidence. How many randomly selected professional employees would they need
to sample?
c. Suppose 40 professional employees are randomly selected. This sample yields a mean
of 15.2 hours and a standard deviation of 1.8 hours. Find a 95% confidence interval and
interpret.

Accepted Answer

The answer of A large software development firm recently relocated...

Question 7

Marcy's Consignment shop is based in Port Angeles, Washington. It offers both male and female never or slightly used clothing and accessories on a consignment basis.
Marcy's recently redesigned their website and wants to show that sales have increased.
The sales manager selects 30 sales records at random from the store's online client list
And she finds a mean increase in spending of $3.50 with a standard deviation of $7.20.
The resulting P-value for anα of 0.05 is

Accepted Answer

A)   0.012
B)   0.315
C)   0.630
D)   0.006
E)   0.024

Question 8

A pharmaceutical company wants to answer the question whether it takes longer than 45 seconds for a drug in pill form to dissolve in the gastric juices of the stomach. A sample was taken from patients taken the given drug in pill form and times for the pills to be dissolved were measured as the following: $$\begin{array} { | l | l | l | l | l | l | l | l | } 
\hline 42.7 & 43.4 & 44.6 & 45.1 & 45.6 & 45.9 & 46.8 & 47.6 \
\hline
\end{array}$$ 
a. State the hypotheses to test this question.
b. State the assumptions of the test.
c. Perform the test and determine if it is statistically significant at α = 0.05.
d. Interpret the result.

Accepted Answer

The answer of A pharmaceutical company wants to answer the...

Question 9

In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist
Collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and
A standard deviation of 0.65 feet. Which of the following statements is true?

Accepted Answer

A)   The sampling distribution for the sample mean follows the t-distribution with 16Degrees of freedom.
B)   The sampling distribution for the sample mean follows the t-distribution with 15Degrees of freedom.
C)   The mean of the sampling distribution for the sample mean is 14.8 feet.
D)   The sampling distribution for the sample mean is Normal with a mean of 14.8 feetAnd standard deviation of .65 feet.
E)   The standard error is 0.65 feet.

Question 10

A small business ships specialty homemade candies to anywhere in the world. Past
records indicate that the weight of orders is normally distributed. Suppose a random
sample of 16 orders is selected and each is weighed. The sample mean was found to be
110 grams with a standard deviation of 14 grams.
A. Describe the sampling distribution for the sample mean.
b. What is the standard error?
c. For 90% confidence, what is the margin of error?
d. Based on the sample results, create the 90% confidence interval and interpret.

Accepted Answer

The answer of A small business ships specialty homemade candies...

Question 11

A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random
Sample of 16 orders is selected and each is weighed. The sample mean was found to be
110 grams with a standard deviation of 14 grams. Which of the following statements is
True?

Accepted Answer

A)   The sampling distribution for the sample mean follows the t-distribution with 16Degrees of freedom.
B)   The sampling distribution for the sample mean follows the t-distribution with 15Degrees of freedom.
C)   The mean of the sampling distribution for the sample mean is 110 grams.
D)   The sampling distribution for the sample mean is Normal with a mean of 110 gramsAnd standard deviation of 14 grams.
E)   The standard error is 14 grams.

Question 12

For a one-sided alternative hypothesis, the critical t value for an α of 0.02 and df of 19 is

Accepted Answer

A)   1.734
B)   2.093
C)   2.540
D)   1.729
E)   3.883

Question 13

A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the
Ravioli is normally distributed. To make sure that the automatic filling process is on
Target, quality control inspectors take a sample of 25 ravioli. The correct value of t* to
Construct a 98% confidence interval for the true mean amount of cheese filling is

Accepted Answer

A)   2.492
B)   3.467
C)   2.797
D)   2.064
E)   3.745

Question 14

Grandma Gertrude's Chocolates, a family owned business, has an opportunity to supply its product for distribution through a large coffee house chain. However, the
Coffee house chain has certain specifications regarding cacao content as it wishes to
Advertise the health benefits (antioxidants) of the chocolate products it sells. In order to
Determine the mean % cacao in its dark chocolate products, quality inspectors sample 36
Pieces. They find a sample mean of 55% with a standard deviation of 4%.
The 90% confidence interval for the true mean % cacao is

Accepted Answer

A)   53.87% to 56.13%.
B)   51% to 59%.
C)   53.33% to 56.67%.
D)   53.64% to 56.36%
E)   54.33% to 55.67%.

Question 15

A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on
data collected from its automatic filling process, the amount of cheese inserted into the
ravioli is normally distributed. To make sure that the automatic filling process is on
target, quality control inspectors take a sample of 25 ravioli and measure the weight of
cheese filling. They find a sample mean weight of 15 grams with a standard deviation of
1.5 grams.
A. Describe the sampling distribution for the sample mean.
b. What is the standard error?
c. What is the margin of error for 99% confidence?
d. What is the margin of error for 90% confidence?
e. Based on the sample results, find the 99% confidence interval and interpret.
f. Based on the sample results, find the 90% confidence interval and interpret.
g. For a more accurate determination of the mean weight, the quality control inspectors
wish to estimate it within 0.25 grams with 99% confidence. How many ravioli should
they sample?

Accepted Answer

The answer of A manufacturer of cheese filled ravioli supplies...

Question 16

A manufacturer of cheese filled ravioli supplies a pizza restaurant chain. Based on data collected from its automatic filling process, the amount of cheese inserted into the
Ravioli is normally distributed. To make sure that the automatic filling process is on
Target, quality control inspectors take a sample of 25 ravioli and measure the weight of
Cheese filling. They find the 99% confidence interval of 14.16 to 15.84 grams. Which of
The following is the correct interpretation?

Accepted Answer

A)   We are 99% confident that the mean weight of cheese filling in all ravioli made by thisProcess is between 14.16 and 15.84 grams.
B)   99% of all ravioli made by this process will have cheese filling weights between 14.16And 15.84 grams.
C)   The weight of cheese filling in the ravioli is between 14.16 and 15.84 grams 99% ofThe time.
D)   All of the above.
E)   None of the above.

Question 17

A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random
Sample of 16 orders is selected and each is weighed. The sample mean was found to be
110 grams with a standard deviation of 14 grams. The standard error of the mean is

Accepted Answer

A)   14 grams
B)   0.875 grams
C)   .35 grams
D)   3.5 grams
E)   27.5 grams

Question 18

In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist
Collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and
A standard deviation of 0.65 feet. The standard error of the mean is

Accepted Answer

A)   0.65 ft.
B)   0.346 ft.
C)   0.1625 ft.
D)   0.0098 ft.
E)   1.625 ft.

Question 19

Insurance companies track life expectancy information to assist in determining the
cost of life insurance policies. Last year the average life expectancy of all policyholders
was 77 years. ABI Insurance wants to determine if their clients now have a longer life
expectancy, on average, so they randomly sample some of their recently paid policies.
The ages of the clients in the sample are shown below. $$\begin{array} { | l | l | l | l | l | l | l | l | l | l | } 
\hline 86 & 75 & 83 & 84 & 81 & 77 & 78 & 79 & 79 & 81 \
\hline 76 & 85 & 70 & 76 & 79 & 81 & 73 & 74 & 72 & 83 \
\hline
\end{array}$$ 
A. Based on the sample results, find the 90% confidence interval and interpret.
b. For more accurate cost determination, ABI Insurance wants to estimate the average life
expectancy to within one year with 95% confidence. How many randomly selected
recently paid policies would they need to sample?
c. Suppose ABI samples 100 recently paid policies. This sample yields a mean of 77.7
years and a standard deviation of 3.6 years. Find a 90% confidence interval and interpret.

Accepted Answer

The answer of Insurance companies track life expectancy information to...

Question 20

A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random
Sample of 16 orders is selected and each is weighed. The sample mean was found to be
110 grams with a standard deviation of 14 grams. The 90% confidence interval for the
True mean weight of orders is

Accepted Answer

A)   103.87 to 116.14 grams.
B)   86.046 to 133.954 grams.
C)   99.5 to 120.5 grams.
D)   102.55 to 117.45 grams.
E)   103.14 to 116.86 grams.

Exam 11: Confidence Intervals and Hypothesis Tests for Means

For a one-sided alternative hypothesis, the critical t value for an α of 0.02 and df of 19 is