Deck 8: Tests of Hypotheses Based on a Single Sample

Question 1

The rejection region is called __________ if it consists only of small values of the test statistic.

Answer 1

lower-tailed

Question 2

A __________ error is usually more serious than a __________ error.

Answer 2

type I, type II

Question 3

Suppose a test procedure about the population mean is performed, when the population is normal with known standard deviation then if the alternative hypothesis is the rejection region for a level test is __________.

Answer 3

Question 4

A __________ error involves not rejecting the null hypothesis is false.

Question 5

The rejection region is called __________ if it consists only of large values of the test statistic.

Question 6

The null hypothesis will be rejected if and only if the observed or computed __________ value falls in the __________.

Question 7

Suppose a test procedure about the population mean is performed, when the population is normal with known standard deviation then if the alternative hypothesis is the rejection region for a level test is either __________ or __________.

Question 8

If the null hypothesis is then the test statistic value is z = __________.

Question 9

An engineer has suggested a change in the production process in the belief that it will result in a reduced defective rate. Let p denote the true proportion of defective items resulting from the changed process, and that 5% of items produced by a manufacturer during a certain period were defective. Then the research hypothesis is the assertion that __________.

Question 10

The __________ hypothesis should be identified with the hypothesis of no change, no difference, no improvement, and son on.

Question 11

A __________ error consists of rejecting the null hypothesis is true.

Question 12

For any significance level the two-tailed rejection region has type I error probability equal to __________.

Question 13

If is a random sample from a normal distribution, and the sample size n is small, then the standardized variable has a __________ distribution with __________ degrees of freedom.

Question 14

Suppose a test procedure about the population mean is performed, when the population is normal with known standard deviation then if the alternative hypothesis is the rejection region for a level test is __________.

Question 15

The probabilities of type I and type II errors are traditionally denoted by the Greek letters __________ and __________, respectively.

Question 16

Suppose a test procedure about the population mean is performed, when the population is normal and the sample size n is small, then if the alternative hypothesis is the rejection region for a level test is __________.

Question 17

Suppose a test procedure about the population mean is performed, when the population is normal and the sample size n is small, then if the alternative hypothesis is the rejection region for a level test is __________.

Question 18

In many situations, the alternative hypothesis is referred to as the __________ hypothesis, since it is the statement the researcher would really like to validate.

Question 19

The value that represents the probability of type I error is often referred to as the __________ of the test.

Question 20

In our treatment of hypothesis testing, the __________ hypothesis will always be stated as an equality claim.

Question 21

For an upper-tailed z test, the level of significance is just the area under the z curve to the __________ of the critical value __________.

Question 22

Suppose that a test procedure about the population proportion p is performed, and that the sample proportion is approximately normally distributed. If the alternative hypothesis is , then the rejection region for a level test is __________.

Question 23

The smallest level of significance for which the null hypothesis would be rejected is the tail area captured by the computed value of the test statistic. This smallest is referred to as the __________.

Question 24

If the calculated test statistic for an upper-tailed z test is 2.15, then the P-value is __________.

Question 25

Suppose that a test procedure about the population proportion p is performed, and that the sample proportion is approximately normally distributed. If the alternative hypothesis is then the rejection region for a level test is __________.

Question 26

If the calculated test statistic for two-tailed z test is -1.84, then the P-value is __________.

Question 27

Suppose that a t test of is based on 18 degrees of freedom. If the calculated value of the test statistic is t = - 1.80, then the P-value for this test is __________.

Question 28

If the P-value is smaller than or equal to the level of significance then the researcher should __________ at level

Question 29

Suppose that a test procedure about the population proportion p is performed, and that the sample proportion is approximately normally distributed. If the alternative hypothesis is , then the rejection region for a level test is either __________ or __________.

Question 30

Suppose that a z test of is conducted. The critical value is determined by specifying __________ and using the fact that has a __________ distribution when is true.

Question 31

Let p denote the proportion of individuals in a population who possess a specified property, and X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is small relative to the population size, then X has approximately a __________ distribution.

Question 32

Suppose that when data from an experiment was analyzed, the P-value for testing was calculated as .047. At the .01 level, would __________.

Question 33

Suppose that a z test of is conducted. Intuition then suggests rejecting when the value of test statistic z is __________.

Question 34

Suppose a test procedure about the population mean is performed, when the population is normal and the sample size n is small, then if the alternative hypothesis is the rejection region for a level test is either __________ or __________.

Question 35

It is customary to call the data significant when the null hypothesis is __________, and not significant otherwise.

Question 36

Let p denote the proportion of individuals in a population who possess a specified property, and X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is large, then both X and the estimator are approximately __________ distributed.

Question 37

Let p denote the proportion of individuals in a population who possess a specified property, and X denote the number of individuals in the sample who possess the same property. The estimator is __________ if and its standard deviation = __________.

Question 38

As the level of significance decreases, the critical value __________.

Question 39

If the P-value is larger than the level of significance then the researcher should __________ at level

Question 40

Suppose that a t test of is based on 10 degrees of freedom. If the calculated value of the test statistic is t = 2.4, then the P-value for this test is __________.

Question 41

Which of the following statements are not true?

A) A test statistic is a function of the sample data on which the decision to reject or not reject the null hypothesis is to be based.
B) A rejection region consists of the set of all test statistic values for which the null hypothesis will be rejected.
C) A rejection region consists of the set of all test statistic values for which the alternative hypothesis will be rejected.
D) A good hypothesis-testing procedure is one for which the probability of making either type I or type II error is small.
E) None of the above statements are true.

Question 42

Which of the following statements are true?

A) When the alternative hypothesis is $H _ { \pm } : \mu < \mu _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\bar { x }$
Is too far to the left of $\mu _ { 0 }$
)
B) When the alternative hypothesis is $H _ { \pm } : \mu > \mu _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\bar { x }$
Is too far to the right of $\mu _ { 0 }$
)
C) When the alternative hypothesis is $H _ { \pm } : \mu \neq \mu _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\bar { x }$
Is too far to either side of $\mu _ { 0 }$
)
D) All of the above statements are true.
E) None of the above statements are true.

Question 43

In hypothesis-testing analysis, a type II error occurs only if

A) the null hypothesis is rejected when it is true.
B) the null hypothesis is rejected when it is false.
C) the null hypothesis is not rejected when it is false.
D) the null hypothesis is not rejected when it is true.

Question 44

Which of the following statements are true in testing $H _ { 0 } : \mu = 275 \text { versus } H _ { \pm } : > 275$ based on a sample of size 15 from a normal population with unknown standard deviation $\sigma$ ?

A) The test procedure requires the use of standard normal distribution.
B) The test procedure requires the use of binomial distribution.
C) The test procedure requires the use of exponential distribution.
D) The test procedure requires the use of t-distribution with 15 degrees of freedom.
E) None of the above statements are true.

Question 45

If $\theta$ denotes the parameter of interest, and the simplified null hypothesis has the form $H _ { o } : \theta = \theta _ { 0 } , \text { where } \theta _ { o }$ is a specified number called the "null value" of the parameter, then the alternative hypothesis will be

A) $H _ { \pm } : \theta > \theta _ { o }$
(so the implicit null hypothesis is $\left. \theta \leq \theta _ { o } \right)$
B) $H _ { \pm } : \theta < \theta _ { o }$
(so the implicit null hypothesis is $\left. \theta \geq \theta _ { o } \right)$
C) $H _ { \pm } : \theta \neq \theta _ { o }$
D) The alternative hypothesis will look like any of the above three assertions.
E) The alternative hypothesis must be the assertion specified in (C).

Question 46

Which of the following statements are not true if a test procedure about the population mean $\mu$ is performed when the population is normal with known standard deviation $\sigma$ ?

A) The rejection region for level $\alpha$
Test is $z \geq z _ { \alpha}$
If the test is an upper-tailed test.
B) The rejection region for level $\alpha$
Test is $z \geq z _ { \alpha / 2 }$
If the test is an upper-tailed test.
C) The rejection region for level $\alpha$
Test is $z \leq - z _ { \alpha }$
If the test is a lower-tailed test.
D) The rejection region for level $\alpha$
Test is either $z \geq z _ { \alpha/ 2 }$
Or $z \leq - z _ { \alpha/2 }$
If the test is a two-tailed test.
E) None of the above statements are true.

Question 47

Which of the following statements are true?

A) Knowledge of the test statistic's distribution when the null hypothesis is true allows us to construct a rejection region for which the type I error probability is controlled at the desired level.
B) The rejection region for the t test differs from that for the z test only in that a t critical value replaces the z critical value.
C) If $X _ { 1 } , X _ { 2 } , \ldots \ldots , X _ { n }$
(n is small) is a random sample from a normal distribution, the standardized variable $T = ( \bar { X } - \mu ) / ( S / \sqrt { n } )$
Has a t distribution with n-1 degrees of freedom.
D) All of the above statements are true.
E) None of the above statements are true.

Question 48

One must be especially careful in interpreting evidence when the sample size is __________, since any small departure from will almost surely be detected by a test, yet such a departure may have little practical significance.

Question 49

The likelihood ratio test procedure consists of rejecting the null hypothesis when the likelihood ratio statistic value is __________.

Question 50

Which of the following statements are true?

A) The probability of type I error, $\alpha$
Is computed using the probability distribution of the test statistic when the null hypothesis is true.
B) The probability of type II error, $\beta$
Requires knowing the distribution of the test statistic when the null hypothesis is false.
C) The probability of type I error, $\alpha$
Is computed by summing over probabilities of test statistic values in the rejection region.
D) The probability of type II error, $\beta$
Is computed by summing over probabilities of test statistic values in the complement of the rejection region.
E) All of the above statements are true.

Question 51

Which of the following statements are not true?

A) A rejection region is called upper-tailed if it consists only of large values of the test statistic.
B) A rejection region is called upper-tailed if it consists only of small values of the test statistic.
C) A rejection region is called lower-tailed if it consists only of small values of the test statistic.
D) All of the above statements are true.
E) None of the above statements are true.

Question 52

Which of the following statements are not correctly stated?

A) The two possible conclusions from a hypothesis-testing analysis are rejecting the null hypothesis $H _ { o }$
Or accepting $H _ { o }$
)
B) In many situations, the alternative hypothesis $H _ { \pm }$

Is referred to as the "research hypothesis" since it is the claim that the researcher would really like to validate.
C) In our treatment of hypothesis testing, the null hypothesis will always be stated as an equality claim.
D) A test statistic is a rule, based on sample data, for deciding whether to reject the null hypothesis.
E) All of the above statements are correctly stated.

Question 53

Which of the following statements are not correct?

A) It is possible that the null hypothesis may be rejected when it is true.
B) It is impossible that the null hypothesis may be rejected when it is true.
C) It is possible that the null hypothesis may not be rejected when it is false.
D) All of the above statements are correct.
E) None of the above statements are correct.

Question 54

Which of the following statements are not generally true?

A) A type I error is usually more serious than a type II error.
B) A type II error is usually more serious than a type I error.
C) A test with significance level $\alpha$
Is one for which the type I error probability is controlled at the specified level.
D) When an experiment and a sample size are fixed, then decreasing the size of the rejection region to obtain a smaller value of $\alpha$
(probability of type I error) results in a larger value of $\beta$
(probability of type II error) for any particular parameter value consistent with the alternative hypothesis $H _ { \pm }$
)
E) None of the above statements are true.

Question 55

Which of the following statements are not true?

A) A statistical hypothesis is a claim or assertion either about the value of a single parameter, about the values of several parameters, or about the form of an entire probability distribution.
B) In any hypothesis-testing problem, there are two contradictory hypotheses under consideration.
C) A test of hypothesis is a method for using sample data to decide whether the null hypothesis should be rejected.
D) A type I error consists of not rejecting the null hypothesis $H _ { o } \text { when } H _ { o }$
Is false.
E) None of the above statements are true.

Question 56

Which of the following statements are true?

A) The null hypothesis, denoted by $H _ { o }$
, is the claim that is initially assumed to be true (the "prior belief" claim).
B) The alternative hypothesis, denoted by $H _ { \pm }$
, is the assertion that is contradictory to the null hypothesis $H _ { o }$
)
C) The null hypothesis $H _ { o }$
Will be rejected in favor of the alternative hypothesis only if sample evidence suggests that $H _ { o }$
Is false.
D) If sample evidence does not strongly contradict the null hypothesis $H _ { o }$
, we will continue to believe in the truth of $H _ { o }$
)
E) All of the above statements are true.

Question 57

Which of the following statements are not true for any significance level?

A) The one-tailed rejection region $z \geq z _ { \alpha }$
Has type I error probability $\alpha$
B) The one-tailed rejection region $z \geq - z _ { \alpha }$
Has type I error probability $\alpha$
C) The two-tailed rejection region $z \geq z _ { \alpha/ 2 } \text { or } z \leq - z _ { \alpha / 2 }$
Has type I error probability $\alpha / 2$
, since area $\alpha / 2$
Is captured under each of the tow tails of the z curve.
D) All of the above statements are true.
E) None of the above statements are true.

Question 58

In hypothesis-testing analysis, a type I error occurs only if

A) the null hypothesis is rejected when it is true
B) the null hypothesis is rejected when it is false
C) the null hypothesis is not rejected when it is false
D) the null hypothesis is not rejected when it is true

Question 59

Suppose that a two-tailed test procedure about the population mean $\mu$ is performed when the population is normal, but the sample size n is small. The null hypothesis will be rejected at significance level $\alpha$ if the value of the standardized test statistic t is such that

A) $t \geq t _ { \alpha , n - 1 }$
B) $t \leq - t _ { \alpha _ { i } , n- 1 }$
C) either $t \geq t _ { \alpha / 2 , n - 1 } \text { or } t \leq - t _ { \alpha / 2 , n - 1 }$
D) $- t _ { \alpha/ 2 , n - 1 } \leq t \leq t _ { \alpha/2, n - 1 }$
E) None of the above inequalities are correct.

Question 60

Which of the following statements are not true?

A) If the null hypothesis is $H _ { o } : \mu = 50 , \bar { x } = 53 \text {, and } \sigma _ { \bar { x } } = \sigma / \sqrt { n } = 1.2$
, then the test statistic value is t = -2.5.
B) If the alternative hypothesis has the form $H _ { \pm } : \mu > \mu _ { o }$
, then an $\bar { x }$
Value less than $\mu _ { 0 }$
Certainly does not provide support for $H _ { \pm }$
)
C) If the alternative hypothesis has the form $H _ { \pm } : \mu > \mu _ { o }$
, then an $\bar { x }$
Value that exceeds $\mu _ { 0 }$
By only a small amount (corresponding to z which is positive but small) does not suggest that $H _ { o }$
Should be rejected in favor of $H _ {\pm }$
)
D) All of the above statements are true.
E) None of the above statements are true.

Question 61

Which of the following statements are needed in constructing the likelihood ratio test?

A) Finding the largest value of the likelihood for any $\theta \text { in } \Omega _ { 0 }$
(by finding the maximum
Likelihood estimate within $\Omega _ { 0 }$
And substituting back into the likelihood function).
B) Find the largest value of the likelihood for any $\theta \text { in } \Omega _ { \mathrm { 0 } }$
C) Forming the ratio $\lambda \left( x _ { 1 } , \ldots , \ldots , x _ { n } \right)$
= maximum likelihood for $\theta \text { in } \Omega _ { 0 } /$
Maximum likelihood for $\theta \text { in } \Omega _ { \mathrm { 0 } }$
D) All of the above statements are needed.
E) Only statements A and B are needed.

Question 62

Which of the following statements are not true if a test procedure about the population proportion p is performed provided that $np _ { { o } } \geq 10 \text { and } n \left( 1 - p _ { 0 } \right) \geq 10 ?$

A) The rejection region for level $\alpha$
Test is $z \geq z _ { \alpha / 2 }$
If the test is an upper-tailed test.
B) The rejection region for level $\alpha$
Test is $z \leq - z _ { \alpha }$
If the test is a lower-tailed test.
C) The rejection region for level $\alpha$
Test is either $z \geq z _ { \alpha / 2 } \text { or } z \leq - z _ { \alpha / 2 }$
If the test is a two-tailed test.
D) The rejection region for level $\alpha$
Test is $z \geq z _ { \alpha }$
If the test statistic is an upper-tailed test.
E) None of the above statements are true.

Question 63

Let p denote the proportion of individuals in a population who possess a specified property, and let X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is large, the estimator $\hat { p } = \bar{X} / n$ has approximately

A) a normal distribution
B) a binomial distribution
C) an exponential distribution
D) a Poisson distribution

Question 64

Which of the following statements are not true?

A) If the alternative hypothesis has the form $H _ { ± } : p > p _ { o }$
, then a $\hat { p } \text { value less than } p _ { o }$
Certainly does not provide support for $H _ { \pm}$
)
B) If the alternative hypothesis has the form $H _ { ± } : p > p _ { o }$
, then a $\hat { p } \text { value that ex ceeds } p _ { o }$
By only a small amount (corresponding to z value which is positive but small) does not suggest that $H _ { o }$
Should be rejected in favor of $H _ { \pm }$
)
C) If the null hypothesis is $H _ { o } : p = .50 , \hat { p } = .51$

And n = 100, then the test statistic value is z = 4
D) All of the above statements are true.
E) None of the above statements are true.

Question 65

Let denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing versus Explain your reasoning. (Hint: Think about the consequences of a type I and type II error for each possibility.)

Question 66

Which of the following statements are necessary to construct an appropriate test procedure?

A) Specify a test statistic.
B) Decide on the general form of the rejection region.
C) Select the specified numerical critical value or values that will separate the rejection region from the acceptance region.
D) All of the above statements are necessary.
E) Only A and B are necessary statements.

Question 67

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05mm. What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

Question 68

Which of the following statements are true about the P-value, where z is the calculated value of the test statistic and $\Phi ( z )$ is the corresponding cumulative area under the standard normal curve?

A) P-value = $1 - \Phi ( z )$
For an upper-tailed test.
B) P-value = $\Phi ( z )$
For a lower-tailed test.
C) P-value = $2 [ 1 - \Phi ( \mid z ) ]$
For a lower-tailed test
D) All of the above statements are true.
E) None of the above statements are true.

Question 69

Let the test statistic Z have a standard normal distribution when is true. Give the significance level for each of the following situations.
a. b. c.

Question 70

Which of the following statements are true?

A) When the alternative hypothesis is $H _ { ± } : p < p _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\hat { p }$
Is too far to the left of $p _ { o }$
)
B) When the alternative hypothesis is $H _ { ± } : p > p _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\hat { p }$
Is too far to the right of $p _ { o }$
)
C) When the alternative hypothesis is $H _ { \pm} : p \neq p _ { o }$
, the null hypothesis $H _ { o }$
Should be rejected if $\hat { p }$
Is too far to either side of $p _ { o }$
)
D) All of the above statements are true
E) None of the above statements are true.

Question 71

Let p denote the proportion of individuals in a population who possess a specified property, and let X denote the number of individuals in the sample who possess the same property. Provided that the sample size n is small relative to the population size, X has approximately

A) a normal distribution
B) a binomial distribution
C) an exponential distribution
D) a Poisson distribution

Question 72

Which of the following statements are true?

A) When the results of an experiment are to be communicated to a large audience, rejection of $H _ { o }$
At level $\alpha$
Will be much more convincing if the observed value of the test statistic greatly exceeds the $\alpha \%$
Critical value than if it barely exceeds that value.
B) A large P-value would indicate statistical significance in that it would strongly suggest rejection of $H _ { o } \text { in favor of } H _ { \pm }$
C) In many experimental situations, only departures from $H _ { o }$
Of small magnitude would be worthy of detection, whereas a large departure from $H _ { o }$
Would have little practical significance.
D) All of the above statements are true.
E) None of the above statements are true.

Question 73

Which of the following statements are not true?

A) A P-value conveys much information about the strength of evidence against the null hypothesis $H _ { o }$
And allows an individual decision maker to draw a conclusion at any specified significance level $\alpha .$
B) The P-value (or observed significance level) is the largest level of significance at which the null hypothesis $H _ { o }$
Would be rejected when a specified test procedure is used on a given data set.
C) If P-value $\leq \alpha , \text { wereject } H _ { o } \text { at level } \alpha \text {. }$
D) If P-value $> \alpha \text {, we do not reject } H _ { o } \text { at level } \alpha \text {. }$
E) All of the above statements are true.

Question 74

Suppose that when data from an experiment was analyzed, the P-value for testing $H _ { o } : \mu = 50 \text { ver sus } H _ {\pm} : \mu > 50$ was calculated as .0244. Which of the following statements are true?

A) $H _ { o }$
Is rejected at .10 level.
B) $H _ { o }$
Is not rejected at .05 level.
C) $H _ { o }$
Is not rejected at .025 level.
D) $H _ { o }$
Is rejected at any level $\alpha .$
E) All of the above statements are true.

Question 75

Which of the following statements are true if the value of the test statistic for a two-tailed z test is z = -1.56?

A) P-value = .4406
B) P-value = .0594
C) P-value = .1188
D) .0594 < P-value < .1188
E) P-value = .0406

Question 76

Which of the following statements are not true?

A) The test statistic $Z = \left( \bar { X } - \mu _ { 0 } \right)/ ( \sigma / \sqrt { n } )$
Has a standard normal distribution when $H _ { o } : \mu = \mu _ { 0 }$
Is true.
B) The reliability of hypothesis testing procedure in reaching a correct decision can be assessed by studying type I error probability.
C) The process of reaching a decision by using the methodology of classical hypothesis testing involves selecting a level of significance $\alpha$
And then rejecting or not rejecting the null hypothesis $H _ { o }$
At that level $\alpha .$
D) All of the above statements are true.
E) None of the above statements are true.

Question 77

Which of the following P-values will lead us to reject the null hypothesis at the .05 level?

A) .10
B) .025
C) .075
D) .15
E) Any P-value greater than .05

Question 78

Suppose that a t test of $H _ { o } : \mu = 250 \text { versus } H _ { \pm } : \mu \neq 250$ is based on 12 degrees of freedom. If the calculated value of the test statistic is 2.8, then the P-value is

A) .008
B) .992
C) .016
D) .492
E) .496

Question 79

Which of the following statements are true?

A) It is customary to call the data significant when $H _ { o }$
Is rejected.
B) It is customary to call the data not significant when $H _ { o }$
Is not rejected.
C) The calculation of the P-value depends on whether the test is upper-tailed, lower-tailed, or two-tailed.
D) The P-value for a z test (one based on a test statistic whose distribution when $H _ { o }$
Is true is at least approximately standard normal) is easily determined from the information in the standard normal probability table.
E) All of the above are true statements.

Question 80

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X.
a. Which of the following rejection regions is most appropriate and why?
or
b. In the context of this problem situation, describe what type I and type II errors are.
c. What is the probability distribution of the test statistic X when
is true? Use it to compute
the probability of a type I error..
d. Compute the probability of a type II error for the selected region when p = .6 and p = .7.
e. Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?