Deck 7: Statistical Intervals Based on a Single Sample

Question 1

The formula used to construct approximately confidence interval for a population proportion p when the sample size n is large enough is given by __________, where is the sample proportion, and

Answer 1

Question 2

The chi-squared critical value, , denotes the number on the measurement axis such that __________ of the area under the chi-squared curve with __________ degrees of freedom lies to the __________ of .

Answer 2

right

Question 3

Let be a random sample from a normal distribution with mean and variance . Then the random variable has a __________ probability distribution with __________ degrees of freedom.

Answer 3

chi-squared

Question 4

The area under a t-density curve between the critical values is __________.

Question 5

Let be a random sample from a population having a mean and standard deviation . Provided that n is large, the Central Limit Theorem (CLT) implies that is __________ distributed.

Question 6

When is the mean of a random sample of size n (n is small) from a normal population with mean , the random variable has a probability distribution called t-distribution with n-1 __________.

Question 7

A random sample of 50 observations produced a mean value of 55 and standard deviation of 6.25. The 95% confidence interval for the population mean is between __________ and __________. (two decimal places)

Question 8

A large-sample lower confidence bound for the population mean __________.

Question 9

The z curve is often called the t curve with degrees of freedom equal to __________.

Question 10

If we think of the width of the confidence interval as specifying its precision or accuracy, then the confidence level (or reliability) of the interval is __________ related to its precision.

Question 11

If you want to develop a 99% confidence interval for the mean of a normal population, when the standard deviation is known, the confidence level is __________.

Question 12

The standard normal random variable has a mean value of __________ and standard deviation of __________.

Question 13

If a confidence level of 90% is used to construct a confidence interval for the mean of a normal population when the value of the standard deviation is known, the z critical value is __________.

Question 14

The ability of a confidence interval to contain the value of the population mean is described by the __________.

Question 15

The 90^th percentile of a chi-squared distribution with 15 degrees of freedom is equal to __________.

Question 16

If the random sample is taken from a normal distribution with mean value and standard deviation , then regardless of the sample size n, the sample mean is distributed with expected value __________ and standard deviation __________.

Question 17

When is the mean of a random sample of size n (n is large) from a normal population with mean , the random variable has approximately a __________ distribution with mean value of __________ and standard deviation of __________.

Question 18

The 5^th percentile of a chi-squared distribution with 10 degrees of freedom is equal to __________.

Question 19

The formula used to construct a 95% confidence interval for the mean of a normal population when the value of the standard deviation is known is given by __________.

Question 20

Let denote the density function curve for a t-distribution with degrees of freedom. As __________, the spread of the corresponding curve decreases.

Question 21

Which of the following statements are not true?

A) Provided that the sample size n is large, the standardized variable $Z = ( \bar { X } - \mu ) / ( \sigma / \sqrt { n } )$
Is approximately normally distributed, while the variable $Z = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$
Is not.
B) The formula $\bar { x } \pm z _ { \alpha/2 } \cdot s / \sqrt { n }$
Is a large-sample confidence interval for $\mu$
With confidence level approximately $100 ( 1 - \alpha ) \%$
)
C) Generally speaking, n >40 will be sufficient to justify the use of the formula $\bar { x } \pm z _ { \alpha/2 } \cdot s / \sqrt { n }$
As a large-sample confidence interval for $\mu$
)
D) None of the above statements are true.
E) All of the above statements are true.

Question 22

The area under a chi-squared curve with 10 degrees of freedom, which is captured between the two critical values is __________.

Question 23

Which of the following statements are not true?

A) The notation $t _ { \boldsymbol { \alpha } , \boldsymbol { v } }$
Is often used to denote the number on the measurement axis for which the area under the t-curve with \v
Degrees of freedom to the left of $t _ { \boldsymbol { \alpha } , \boldsymbol { v } }$
Is $\alpha$
, where $t _ { \boldsymbol { \alpha } , \boldsymbol { v } }$
Is called a t critical value.
B) The number of degrees of freedom for a t- variable is the number of freely determined deviations $X _ { 2 } - \bar { X }$
On which the estimated standard deviation in the denominator of $T = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$
Is based.
C) A larger value of degrees of freedom $v$
Implies a t-distribution with smaller spread.
D) All of the above statements are true.
E) None of the above statements are true.

Question 24

Which of the following statements are not true if $t _ { v }$ denotes the density function curve for a t-distribution with $v$ degrees of freedom?

A) The t-distribution is governed by $\gamma$
Only.
B) Each $t_{J}$
Curve is bell-shaped and centered around 0.
C) Each $t_{J}$

Curve is less spread out than the standard normal z curve.
D) As $\gamma$
Increases, the spread of the corresponding $t_{J}$

Curve decreases.
E) None of the above answers are true.

Question 25

If the width of a confidence interval for $\mu$ is too wide when the population standard deviation $\sigma$ is known, which one of the following is the best action to reduce the interval width?

A) Increase the confidence level
B) Reduce the population standard deviation $\sigma$
C) Increase the population mean $\mu$
D) Increase the sample size n
E) None of the above answers are correct.

Question 26

Which of the following statements are not true?

A) A correct interpretation of a $100 ( 1 - \alpha ) \%$
Confidence interval for the mean $\mu$
Relies on the long-run frequency interpretation of probability.
B) It is correct to write a statement such as $P [ \mu \text { lies in the interval } ( 70,80 ) ] = .95$
C) The probability is .95 that the random interval $\bar { X } \pm 1.96 \cdot \sigma / \sqrt { n }$
Includes or covers the true value of $\mu$
)
D) The interval $\bar { x } \pm 1.645 \cdot \sigma / \sqrt { n }$
Is a 90% confidence interval for the mean $\mu$
)
E) None of the above statements are true.

Question 27

A random sample of 100 observations produced a sample proportion of .25. An approximate 90% confidence interval for the population proportion p is

A) .248 and .252
B) .179 and .321
C) .423 and .567
D) .246 and .254
E) None of the above answers are correct.

Question 28

Suppose that an investigator believes that virtually all values in the population are between 38 and 70. The appropriate sample size for estimating the true population mean $\mu$ within 2 units with 95% confidence level is approximately

A) 61
B) 62
C) 15
D) 16
E) None of the above answers are correct.

Question 29

Which of the following statements are not true in developing a confidence interval for the population mean $\mu$

A) The width of the confidence interval becomes narrower when the sample mean increases.
B) The width of the confidence interval becomes wider when the sample mean increases.
C) The width of the confidence interval becomes narrower when the sample size n increases.
D) All of the above statements are true.
E) None of the above statements are true.

Question 30

Which of the following statements are true?

A) A confidence interval is always calculated by first selecting a confidence level, which is a measure of the degree of reliability of the interval.
B) A confidence level of 95% implies that 95% of all samples would give an interval that includes the parameter being estimated, and only 5% of all samples would yield an erroneous interval.
C) Information about the precision of an interval estimate is conveyed by the width of the interval.
D) The higher the confidence level, the more strongly we believe that the value of the parameter being estimated lies within the interval.
E) All of the above statements are true.

Question 31

A 99% confidence interval for the population mean $\mu$ is determined to be (65.32 to 73.54). If the confidence level is reduced to 90%, the 90% confidence interval for $\mu$

A) becomes wider
B) becomes narrower
C) remains unchanged
D) None of the above answers are correct.

Question 32

If one wants to develop a 90% confidence interval for the mean $\mu$ of a normal population, when the standard deviation $\sigma$ is known, the confidence level is

A) .10
B) .45
C) .90
D) 1.645

Question 33

In developing a confidence interval for the population mean $\mu$ , a sample of 50 observations was used, and the confidence interval was 15.24 $\pm$ 1.20. Had the sample size been 200 instead of 50, the confidence interval would have been

A) 7.62 $\pm$
1)20
B) 15.24 $\pm$
)30
C) 15.24 $\pm$
)60
D) 3.81 $\pm$
1)20
E) None of the above answers are correct.

Question 34

A random sample of 10 observations was selected from a normal population distribution. The sample mean and sample standard deviations were 20 and 3.2, respectively. A 95% prediction interval for a single observation selected from the same population is

A) 20 $\pm$
6)152
B) 20 $\pm$
4)244
C) 20 $\pm$
7)962
D) 20 $\pm$
7)592
E) None of the above answers are correct.

Question 35

Which of the following statements are true?

A) The interval $\bar { X } \pm 1.96 \cdot \sigma/ \sqrt { n }$
Is random, while its width is not random.
B) The interval $\bar { X } \pm 1.96 \cdot \sigma / \sqrt { n }$
Is not random, while its width is random.
C) The interval $\bar { x } \pm 1.96 \cdot \sigma / \sqrt { n }$
Is random, while its width is not random.
D) The interval $\bar { x } \pm 1.96 \cdot \sigma / \sqrt { n }$
Is not random, while its width is random.
E) None of the above statements are true.

Question 36

A 99% confidence interval for the mean $\mu$ of a normal population when the standard deviation $\sigma$ is known is found to be 98.6 to 118.4. If the confidence level is reduced to .95, the confidence interval for $\mu$

A) becomes wider
B) becomes narrower
C) remains unchanged
D) None of the above answers are correct.

Question 37

Which of the following statements are true when $\bar { X }$ is the mean of a random sample of size n from a normal distribution with mean $\mu$ ?

A) The random variable $Z = ( \bar { X } - \mu ) / ( S / \sqrt { n ) }$
Has approximately a standard normal distribution for large n.
B) The random variable $T = ( \bar { X } - \mu ) / ( S / \sqrt { n ) }$
Has a t-distribution with n-1 degrees of freedom for small n.
C) The normal distribution is governed by two parameters, the mean $\mu$
And the standard deviation $\sigma$
)
D) A t-distribution is governed by only one parameter, called the number of degrees of freedom.
E) All of the above answers are true.

Question 38

A random sample of 64 observations produced a mean value of 82 and standard deviation of 5.5. The 90% confidence interval for the population mean $\mu$ is between

A) 81.86 and 82.14
B) 80.65 and 83.35
C) 80.87 and 83.13
D) 81.31 and 82.69
E) None of the above answers are correct.

Question 39

Which of the following statements are true?

A) The price paid for using a high confidence level to construct a confidence interval is that the interval width becomes wider.
B) The only 100% confidence interval for the mean $\mu$
Is $( - \infty , \infty )$
)
C) If we wish to estimate the mean $\mu$
Of a normal population when the value of the standard deviation $\sigma$
Is known, and be within an amount B with $100 ( 1 - \alpha ) \%$
Confidence, the formula for determining the necessary sample size n is $n = \left[ z _ { \alpha / 2 } \cdot \sigma / B \right] ^ { 2 }$
)
D) All of the above statements are true.
E) None of the above statements are true.

Question 40

Which of the following expressions are true about a large-sample upper confidence bound for the population mean $\mu$ ?

A) $\mu < \bar { x } - z _ { \alpha / 2 } \cdot 5 \sqrt { n }$
B) $\mu < \bar { x } + z _ { \alpha / 2 } \cdot 5\sqrt { n }$
C) $\mu < \bar { x } - z _ {\alpha } \cdot 5 \sqrt { n }$
D) $\mu < \bar { x } + z _ { \alpha } \cdot 5 - \sqrt { n }$
E) None of the above statements are true.

Question 41

The lower limit of a 95% confidence interval for the variance $\sigma ^ { 2 }$ of a normal population using a sample of size n and variance value $s ^ { 2 }$ is given by:

A) $( n - 1 ) s ^ { 2 } / \chi _ { .05,_ { n - 1 } } ^ { 2 }$
B) $( n - 1 ) s ^ { 2 } / \chi _ { .025 , n - 1 } ^ { 2 }$
C) $( n - 1 ) s ^ { 2 } / \chi _ { .95, { n - 1 } } ^ { 2 }$
D) $( n - 1 ) s ^ { 2 } / \chi ^ { 2 } _{.975 , n - 1}$
E) None of the above answers are correct.

Question 42

Which of the following statements are false about the chi-squared distribution with $v$ degrees of freedom?

A) It is a discrete probability distribution with a single parameter $v$
)
B) It is positively skewed (long upper tail)
C) It becomes more symmetric as $v$
Increases.
D) All of the above statements are true.
E) All of the above statements are false.

Question 43

A more extensive tabulation of t critical values than what appears in your text shows that for the t distribution with 20 df, the areas to the right of the values .687, .860, and 1.064 are .25, .20, and .15, respectively. What is the confidence level for each of the following three confidence intervals for the mean of a normal population distribution? Which of the three intervals would you recommend be used, and why?
a.
b.
c.

Question 44

By how much must the sample size n be increased if the width of the CI is to be halved? If the sample size is increased by a factor of 25, what effect will this have on the width of the interval? Justify your assertions.

Question 45

It was reported that, in a sample of 507 adult Americans, only 142 correctly described the Bill of Rights as the first ten amendments to the U.S. Constitution. Calculate a (two-sided) confidence interval using a 99% confidence level for the proportion of all U. S. adults that could give a correct description of the Bill of Rights.

Question 46

A study of the ability of individuals to walk in a straight line reported that accompanying data on cadence (strides per seconds) for a sample of n - 20 randomly selected healthy men: A normal probability plot gives substantial support to the assumption that the population distribution of cadence is approximately normal. A descriptive summary of the data from MINITAB follows.
a. Calculate and interpret a 95% confidence interval for a population mean cadence.
b. Calculate and interpret a 95% prediction interval for the cadence of a single individual randomly selected from this population.
c. Calculate an interval that includes at least 99% of the cadences in the population distribution using a confidence level of 95%.

Question 47

A random sample of 100 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec. Calculate a 99% (two-sided) confidence interval for the true average echo duration , and interpret the resulting interval.

Question 48

A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.50 MPa and a sample standard deviation of .80 MPa.
a. Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress?
b. Calculate and interpret a 95% lower prediction bound for the proportional limit stress of a single joint of this type.

Question 49

The upper limit of a 95% confidence interval for the variance $\sigma ^ { 2 }$ of a normal population using a sample of size n and variance value $s ^ { 2 }$ is given by:

A) $( n - 1 ) s ^ { 2 } / \chi _ { .05 , { n - 1 } } ^ { 2 }$
B) $( n - 1 ) s ^ { 2 } / \chi _ { .015 , n - 1 } ^ { 2 }$
C) $( n - 1 ) s ^ { 2 } / \chi _ {.95 ,{ n - 1 } } ^ { 2 }$
D) $( n - 1 ) s ^ { 2 } / \chi ^ { 2 } _{.975 , n - 1}$
E) None of the above answers are correct.

Question 50

The results of a Wagner turbidity test performed on 15 samples of standard Ottawa testing sand were (in microamperes)
a. Is it plausible that this sample was selected from a normal population distribution?
b. Calculate an upper confidence bound with confidence level 90% for the population standard deviation of turbidity.

Question 51

Determine the following:
a. The 90^th percentile of the chi-squared distribution with
= 12.
b. The 10^th percentile of the chi-squared distribution with
= 12.
c.
where
is a chi-squared rv with
= 22.
d.
where
is a chi-squared rv with
= 25

Question 52

A random sample of n = 8 E-glass fiber test specimens of a certain type yielded a sample mean interfacial shear yield stress of 30.5 and a sample standard deviation of 3.0. Assuming that interfacial shear yield stress is normally distributed, compute a 95% CI for true average stress.

Question 53

A CI is desired for the true average stray-load loss (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with = 3.0.
a. Compute a 95% CI for
when n = 25 and
= 60.
b. Compute a 95% CI for
when n = 100 and
= 60.
c. Compute a 99% CI for
when n = 100 and
= 60.
d. Compute an 82% CI for
when n = 100 and
= 60.
e. How large must n be if the width of the 99% interval for
is to be 1.0?

Question 54

The amount of lateral expansion (mils) was determined for a sample of n = 9 pulsed-power gas metal arc welds used in LNG ship containment tanks. The resulting sample standard deviation was s = 2.80 mils. Assuming normality, derive a 95% CI for and for .

Question 55

Consider the 1000 95% confidence intervals (CI) for that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of ? What is the probability that between 950 and 970 of these intervals contain the corresponding value of ? (Hint: Let Y = the number among the 1000 intervals that contain . What kind of random variable is Y?).

Question 56

Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected, and the alcohol content of each bottle is determined. Let denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (8.0, 9.6).
a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning.
b. Consider the following statement: There is a 95% chance that
is between 8 and 9.6. Is this statement correct? Why or why not?
c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 8.0 and 9.6. Is this statement correct? Why or why not?
d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 955 interval ire repeated 100 times, 95 of the resulting intervals will include
. Is this statement correct? Why or why not?

Question 57

Determine the confidence level for each of the following large-sample one-sided confidence bounds:
a. Upper bound:
b. Lower bound:
c. Upper bound:

Question 58

Which of the following statements are true about the percentiles of a chi-squared distribution with 20 degrees of freedom?

A) The 5^th percentile is 31.410
B) The 95^th percentile is 10.851
C) The 10^th percentile is 12.443
D) The 90^th percentile is 37.566
E) All of the above statements are true.

Question 59

The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poisson distribution with parameter . Use the accompanying data on absences for 50 days to derive a large-sample CI for . [Hint: The mean and variance of a Poisson variable both equal , so has approximately a standard normal distribution. Now proceed as in the derivation of the interval for p by making a probability statement (with probability 1 - ) and solving the resulting inequalities for .