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LOS c. explain a test statistic, Type I and Type II errors, a significance level, and how significance levels are used in hypothesis testing;
Test statistic and significance level.
A test statistic is simply a number, calculated from a sample, whose value, relative to its probability distribution, provides a degree of statistical evidence against the null hypothesis. In many cases, the test statistic will not provide evidence sufficient to justify rejecting the null hypothesis. However, sometimes the evidence will be strong enough so that the null hypothesis is rejected and the alternative hypothesis is accepted instead.

The value of the test statistic is the focal point of assessing the validity of a hypothesis. Typically, the test statistic will be of the general form:

For example, a test statistic for the mean of a distribution (such as the mean monthly return for a stock index) often follows a standard normal distribution. In such a case, the test statistic requires use of the z-test, P(Z <= test statistic = z). This is shown as:

where:
X-bar = sample mean
μ0 = hypothesized value
σ = sample standard deviation
n = sample size

Note that this assumes the population variance (and, therefore, the population standard deviation as well) is unknown, and can only be estimated from the sample data.

There are other probability distributions as well, such as the t-distribution, the chi-square distribution, and the F-distribution. Depending on the characteristics of the population and the sample, a test statistic may follow one of these distributions, which will be discussed later.

After setting up H0 and H1, the next step is to state the level of significance, which is the probability of rejecting the null hypothesis when it is actually true. Alpha (α) is used to represent this probability. The idea behind setting the level of significance is to choose the probability that any decision will be subject to a Type I error. There is no one level of significance that is applied to all studies involving sampling. A decision by the researcher must be made to use the 0.01 level, 0.05 level, 0.10 level, or any other level between 0 and 1. A lower level of significance means that there is a lower probability that a Type I error will be made.

Practice Question 1

A consumer group wants to prove that the average hospital costs are more than $931 per day. The group randomly samples 60 accounts and finds a sample mean of $950. The hypothesis test is at an a-level of 5% and assumes a σ of $50. The critical value is ______ standard deviations.

A. 1.645
B. -1.645
C. 1.28

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Practice Question 2

For the hypothesis test below, the test value is ______.

A. 1.75
B. 1.93
C. -1.75

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Practice Question 3

It is desired to test the claim that a steady diet of wolfbane will cause an 18-year-old lycanthrope werewolf to lose EXACTLY 10 lbs. over 5 months. A random sample of 49 lycanthropes was taken, yielding an average weight loss over 5 months of 12.5 lbs. with S = 7 lbs. Let ALPHA = .02. What is the calculated value suitable for testing the above hypothesis?

A. 12.5
B. 7 * 2.5
C. 2.5

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Practice Question 4

The level of significance is:

I. the probability of rejecting the null hypothesis when the null hypothesis is true.
II. the magnitude of the sample size.
III. symbolized by the Greek letter ALPHA.
IV. Level of confidence = 1 - (level of significance).

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Practice Question 5

The critical value of a test statistic is determined from:

A. calculations from the data.
B. calculations based on many actual repetitions of the same experiment.
C. the sampling distribution of the statistic assuming HA.
D. the sampling distribution of the statistic assuming H0.

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Practice Question 6

In hypothesis testing, what is the function of a critical value that is taken from the tables?

A. It is equal to the calculated statistic from the observed data.
B. It is the point where the decision changes from reject to fail to reject.
C. It is the center of the distribution of X's.

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Practice Question 7

True or False? If False, correct it.

If we would reject a null hypothesis at the 5% level, we would also reject it at the 1% level.

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Practice Question 8

True or false? If false, explain why.

Significance at the α = .001 level means that the null hypothesis is almost certainly false.

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Practice Question 9

True or false? If false, explain why.

Testing at a 5% level of significance means that you only have a 5% chance of rejecting the null hypothesis.

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Practice Question 10

Statistical Significance means that, if an experiment were replicated over and over again:

A. the same results would occur again with certainty
B. the same results would probably occur
C. the same results would certainly not occur again.

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Practice Question 11

Which of the following statements regarding hypothesis testing is false?

A. If the population standard deviation is known, then the standard error of the sample mean is found by dividing the population standard deviation by the square root of "n."
B. The test statistic is also known as the "critical value."
C. Despite the different ways to formulate a hypothesis, the null hypothesis is always tested at the point of equality.

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Practice Question 12

What do we call the value z that is computed using sample data?

A. Test statistic
B. Parameter
C. None of these answers

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Practice Question 13

For the hypothesis test below, the test value is ______.

A. 0.577
B. -0.04
C. -0.577

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Practice Question 14

The larger the critical value on the z-statistic,

A. the easier it is to reject the null hypothesis.
B. the harder it is to reject the null hypothesis.
C. the easier it is to accept the null hypothesis.

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Practice Question 15

To test H(0): μ = 20 vs. H(A): μ ≠ 20, a sample of 400 will be taken from a large population, whose standard deviation is 5. H(0) will be rejected if XBAR ≥ 20.5 or XBAR ≤ 19.5. The level of significance of this test is approximately:

A. 0.05
B. 0.02
C. 0.10

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Practice Question 16

You have set up a test of a suitably phrased null hypothesis. After analyzing the statistical underpinnings of the test and the stringency requirements you have imposed, you have determined that the probability of rejecting the null hypothesis equals 30%. The probability that you will reject the null hypothesis when it is true is equal to 13% and the probability that you will reject the null hypothesis when it is false is equal to 17%. The significance level of your test equals ________.

A. 30%
B. 13%
C. 17%

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Practice Question 17

The significance level in hypothesis testing refers to the probability that we will:

A. Reject the null when it is true.
B. Accept the alternative when it is true.
C. Fail to reject the null when it is false.

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Type I and Type II errors in hypothesis testing.
Because hypothesis tests are heavily dependent on the samples used as "evidence", it is definitely possible, in the case of a bad sample, to make an error in the conclusion of a test.

When a hypothesis is tested, there are four possible outcomes:

  1. Reject the null hypothesis when it is false. This is a correct decision.
  2. Incorrectly reject the null hypothesis when it's correct. This is known as the Type I error. The probability of a Type I error is designated by the Greek letter alpha (α) and is called the Type I error rate.
  3. Don't reject the null hypothesis when it's true. This is a correct decision.
  4. Don't reject the null hypothesis when it's false. This is known as the Type II error. The probability of a Type II error (the Type II error rate) is designated by the Greek letter beta (β).

A Type II error is only an error in the sense that an opportunity to reject the null hypothesis correctly was lost. It is not an error in the sense that an incorrect conclusion was drawn since no conclusion is drawn when the null hypothesis is not rejected. It has nothing to do with α, other than the fact that it moves in the opposite direction to α, that is, the bigger the one, the smaller the other.

A Type I error, on the other hand, is an error in every sense of the word. A conclusion is drawn that the null hypothesis is false when, in fact, it is true. Therefore, Type I errors are generally considered more serious than Type II errors. The probability of a Type I error (α) is called the significance level and is set by the experimenter. For example, a 5% level of significance means that there is a 5% probability of rejecting the null when it is true.

There is a tradeoff between Type I and Type II errors. The more an experimenter protects him or herself against Type I errors by choosing a low level, the greater the chance of a Type II error. Requiring very strong evidence to reject the null hypothesis makes it very unlikely that a true null hypothesis will be rejected. However, it increases the chance that a false null hypothesis will not be rejected, thus lowering power. The Type I error rate is almost always set at 0.05 or at 0.01, the latter being more conservative since it requires stronger evidence to reject the null hypothesis at the 0.01 level then at the 0.05 level.

To reduce the probabilities of both types of errors simultaneously, the sample size n must be increased.

Practice Question 1

For the hypothesis test below where the decision is fail to reject H0, a Type II error could have been committed. A correct discussion of the error would be _______.

A. Going along with the population mean is more than 90 when in fact the population mean is 90 or less.
B. Concluding that the population mean is 90 or less when in fact the population mean is less than 90.
C. Going along with the population mean is 90 or less when in fact the population mean is greater than 90.
D. Concluding that the population mean is 90 or more when in fact the population mean is less than 90.

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Practice Question 2

The Santa Rosa Homeowners are trying to prove that the homes in the area are less than 5 miles from the nearest fire station. The data and hypothesis test is shown below. A Type I error could have been made- a discussion of this error is ______.

A. Going along with the homes in Santa Rosa area are on average 5 or more miles from the nearest fire station when in fact the homes are on average less than 5 miles from the nearest fire station.
B. Going along with the homes in Santa Rosa area are on average less than 5 miles from the nearest fire station when in fact the homes are on average 5 or more miles from the nearest fire station.
C. Concluding that the homes in Santa Rosa area are on average 5 or more miles from the nearest fire station when in fact the homes are on average less than 5 miles from the nearest fire station.
D. Concluding that the homes in Santa Rosa area are on average less than 5 miles from the nearest fire station when in fact the homes are on average 5 or more miles from the nearest fire station.

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Practice Question 3

For the hypothesis test below where the decision is reject Ho, then a ______ error could have occurred.

A. Type I
B. Alpha
C. Type II

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Practice Question 4

For the hypothesis test below where the decision is reject H0, a Type I error could have been committed. A correct discussion of the error would be ______

A. Concluding that the population is 90 or more when in fact the population mean is less than 90.
B. Going along with the population mean is 90 or more when in fact it is grater than 90.
C. Concluding that the population mean is less than 90 when it fact is greater than or equal to 90.

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Practice Question 5

The Santa Rosa Homeowners are trying to prove that the homes in the area are less than 5 miles from the nearest fire station. The data and hypothesis test is shown below. A Type II error could have been made- a discussion of this error is ______.

A. Concluding that the homes in Santa Rosa area are on average 5 or more miles from the nearest fire station when in fact the homes are on average less than 5 miles from the nearest fire station.
B. Concluding that the homes in Santa Rosa area are on average less than 5 miles from the nearest fire station when in fact the homes are on average 5 or more miles from the nearest fire station.
C. Going along with the homes in Santa Rosa area are on average 5 or more miles from the nearest fire station when in fact the homes are on average less than 5 miles from the nearest fire station.

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Practice Question 6

In hypothesis testing, when we reject a false null hypothesis, this is a:

A. Type II error.
B. Type I error.
C. correct action.

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Practice Question 7

A type I error in hypothesis testing is when:

A. we do not reject a true null hypothesis.
B. we do not reject a false null hypothesis.
C. we reject a true null hypothesis.

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Practice Question 8

A correct action in hypothesis testing is when:

I. we do not reject a true null hypothesis.
II. we do reject a true null hypothesis.
III. we reject a false null hypothesis.
IV. we do not reject a false null hypothesis.

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Practice Question 9

Of the following statements:

I. The level of significance of a hypothesis test is the probability of rejecting the null hypothesis when it is actually true.
II. Type II error is failing to reject the null hypothesis when it is actually false.

A. Only I is true
B. Only II is true
C. I and II are true

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Practice Question 10

Of the following statements:

I. The level of significance of a hypothesis test is the probability of rejecting the null hypothesis when it is actually true.
II. Type II error is failing to reject the null hypothesis when it is actually false.

A. Only I is true
B. Only II is true
C. I and II are true

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Practice Question 11

For the hypothesis test below where the decision is fail to reject H0, a Type II error could have been committed. A correct discussion of the error would be ______.

A. Going along with the length of rods produced by Machine A is 90 cm when in fact the length is not equal to 90 cm.
B. Concluding that the length of rods produced by Machine A is 90 cm when in fact the length is not equal to 90 cm.
C. Concluding that the length of rods produced by Machine A is not equal to 90 cm when in fact the length is equal to 90 cm.

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Practice Question 12

For the hypothesis test below, a Type I error could have been made. A correct explanation of the error is:

A. Concluding that 70% or less of adults feel that vitamin supplements can prevent illness when in fact more than 70% feel that vitamin supplements can prevent illness.
B. Going along with more than 70% of adults feel that vitamin supplements can prevent illness when in fact 70% or less feel that vitamin supplements can prevent illness.
C. Concluding that more than 70% of adults feel that vitamin supplements can prevent illness when in fact 70% or less feel that vitamin supplements can prevent illness.

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Practice Question 13

If a researcher finds the null hypothesis to be false when it is actually true, a ______ error is occurring?

A. type II
B. type I
C. None of the choices are correct.

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Practice Question 14

If a researcher testing the null hypothesis H0: 53 < μ < 57 vs. Ha: μ < 53 or μ > 57, found that is equal to 56 when in reality is 59, identify the type of error that has been made.

A. Type II
B. Type I
C. None of the choices are correct.

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Practice Question 15

As the alternative mean approaches the hypothesized mean, what can we say about the risk?

A. Smaller risk of a Type II error
B. Greater risk of a Type II error
C. Smaller risk of a Type I error

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Practice Question 16

What is the level of significance?

A. Z-vale of 1.96
B. Beta error
C. Probability of a Type I error

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