A manufacturer claims its batteries last an average of at least 500 hours. A consumer group suspects the true mean life is lower. Which alternative hypothesis fits the consumer group's test?
Ha: mu < 500
✓
Ha: mu > 500
Ha: mu = 500
Ha: x-bar < 500
Correct Answer
Ha: mu < 500
The consumer group wants evidence that the population mean is below 500, so the alternative uses a less-than sign with the parameter mu. Ha: mu > 500 points in the wrong direction for a suspicion that batteries fall short. Ha: mu = 500 cannot be an alternative, because the alternative never contains an equal sign. Ha: x-bar < 500 is a statement about the sample mean, but hypotheses are always about population parameters.
Category: Setting Up Hypotheses
Question 2
A school reports that 40% of its students commute. A researcher believes the true share is different from 40%. Which pair of hypotheses fits this test?
H0: p = 0.40, Ha: p > 0.40
H0: p = 0.40, Ha: p not equal to 0.40
✓
H0: p not equal to 0.40, Ha: p = 0.40
H0: mu = 0.40, Ha: mu not equal to 0.40
Correct Answer
H0: p = 0.40, Ha: p not equal to 0.40
The researcher suspects a difference in either direction, so the alternative says the population proportion p is not equal to 0.40, and the null keeps the reported value. H0: p = 0.40, Ha: p > 0.40 tests only for an increase. H0: p not equal to 0.40, Ha: p = 0.40 reverses the roles, putting the equality in the alternative. H0: mu = 0.40, Ha: mu not equal to 0.40 uses the symbol for a mean, but the claim is about a proportion.
Category: Setting Up Hypotheses
Question 3
An alternative hypothesis is written as Ha: mu not equal to 12. What kind of test does this set up?
Left-tailed test
Right-tailed test
Two-tailed test
✓
Paired-difference test
Correct Answer
Two-tailed test
A not-equal alternative allows the mean to be either above or below 12, so evidence can come from both tails and the test is two-tailed. A left-tailed test comes from an alternative with a less-than sign. A right-tailed test comes from an alternative with a greater-than sign. A paired-difference test is about two matched measurements, not about a single mean compared with 12.
Category: Setting Up Hypotheses
Question 4
Researchers test H0: a new drug is no more effective than a placebo. They conclude the drug is more effective, but in reality it is not. What has happened?
Type II error
Correct rejection of H0
Correct failure to reject H0
Type I error
✓
Correct Answer
Type I error
A Type I error is rejecting a null hypothesis that is actually true, which is what happened when the researchers declared an ineffective drug effective. A Type II error would be failing to reject a false null hypothesis. A correct rejection of H0 would require the drug to really be more effective. A correct failure to reject H0 would mean the researchers kept H0, but they rejected it.
Category: Type I and II Errors
Question 5
A factory tests H0: its machine fills bottles with a mean of 500 mL. The machine is actually overfilling, but the test does not lead to rejecting H0. What has happened?
Type II error
✓
Type I error
Correct rejection of H0
Correct failure to reject H0
Correct Answer
Type II error
A Type II error is failing to reject a null hypothesis that is false, which is what happened when the overfilling machine went undetected. A Type I error would require rejecting a true null hypothesis. A correct rejection of H0 did not happen, because H0 was not rejected. A correct failure to reject H0 would require the machine to be filling correctly, but it was not.
Category: Type I and II Errors
Question 6
In a hypothesis test, what does the significance level alpha represent?
The probability of not rejecting H0 when H0 is false
The probability of rejecting H0 when H0 is true
✓
The probability that H0 is true
The probability of rejecting H0 when H0 is false
Correct Answer
The probability of rejecting H0 when H0 is true
Alpha is the probability of a Type I error, that is, of rejecting a null hypothesis that is actually true. The probability of not rejecting H0 when H0 is false is beta, the chance of a Type II error. The probability that H0 is true is not something a hypothesis test gives, because H0 is either true or false. The probability of rejecting H0 when H0 is false is the power of the test.
Category: Type I and II Errors
Question 7
A test has alpha = 0.05 and beta = 0.15. What is the power of the test?
0.95
0.15
0.85
✓
0.80
Correct Answer
0.85
Power is the probability of correctly rejecting a false null hypothesis, and it equals 1 - beta = 1 - 0.15 = 0.85. The value 0.95 is 1 - alpha, which has nothing to do with power. The value 0.15 is beta itself, the chance of missing a false H0. The value 0.80 comes from subtracting both alpha and beta from 1, which is not a valid formula.
Category: Type I and II Errors
Question 8
Holding the sample size fixed, a researcher lowers alpha from 0.05 to 0.01. What happens to beta?
It decreases
It stays the same
It becomes zero
It increases
✓
Correct Answer
It increases
Making it harder to reject H0 protects against Type I errors, but it also makes it more likely that a false H0 goes unrejected, so beta rises when the sample size does not change. It decreases is backward, because a stricter cutoff rejects less often. It stays the same ignores the trade-off between the two error types. It becomes zero is impossible, because some chance of a Type II error remains for any real test.
Category: Type I and II Errors
Question 9
Which change, with everything else held the same, increases the power of a hypothesis test?
Increasing the sample size
✓
Decreasing the sample size
Lowering alpha from 0.05 to 0.01
Using data with more variability
Correct Answer
Increasing the sample size
A larger sample shrinks the standard error, so a real difference from H0 is easier to detect and the power goes up. Decreasing the sample size enlarges the standard error and lowers power. Lowering alpha from 0.05 to 0.01 makes rejection harder, which lowers power. Using data with more variability also enlarges the standard error and lowers power.
Category: Type I and II Errors
Question 10
A researcher tests a claim about one population mean using a random sample of 15 values from a normal population. The population standard deviation is unknown. Which distribution should the test statistic follow?
Standard normal distribution
Student's t distribution
✓
Binomial distribution
Uniform distribution
Correct Answer
Student's t distribution
When sigma is unknown and is estimated by the sample standard deviation, the standardized sample mean follows Student's t distribution, here with 15 - 1 = 14 degrees of freedom. The standard normal distribution applies to a mean only when sigma is known. The binomial distribution describes counts of successes, not a sample mean. The uniform distribution is not the sampling distribution of a mean.
Category: Choosing the Distribution
Question 11
A one-sample t test of a mean uses a random sample of 25 observations. How many degrees of freedom does the test have?
25
26
24
✓
23
Correct Answer
24
A one-sample t test has n - 1 degrees of freedom, so 25 - 1 = 24. The value 25 uses the sample size without subtracting 1. The value 26 adds 1 instead of subtracting it. The value 23 subtracts 2, which is not the rule for a single sample.
Category: Choosing the Distribution
Question 12
A survey tests a claim about the percentage of adults who own a bicycle, using a large random sample. Which distribution is used for the test statistic?
Student's t distribution
Uniform distribution
Exponential distribution
Normal distribution
✓
Correct Answer
Normal distribution
A test of a single proportion uses the normal approximation to the binomial, so the test statistic is a z-score from the normal distribution. Student's t distribution is used for means when sigma is unknown, not for proportions. The uniform distribution does not describe a sample proportion. The exponential distribution models waiting times, not proportions.
Category: Choosing the Distribution
Question 13
A test gives a p-value of 0.03. Which statement correctly interprets this p-value?
If H0 were true, a result at least this extreme would occur about 3% of the time
✓
There is a 3% chance that H0 is true
There is a 97% chance that Ha is true
There is a 3% chance of making a Type II error
Correct Answer
If H0 were true, a result at least this extreme would occur about 3% of the time
A p-value is computed by assuming H0 is true and finding the probability of a sample result at least as extreme as the one observed. There is a 3% chance that H0 is true is a common misreading, because the p-value assumes H0 rather than measuring its probability. There is a 97% chance that Ha is true makes the same mistake for Ha. There is a 3% chance of making a Type II error confuses the p-value with beta.
Category: P-Values and Decisions
Question 14
A test is run at alpha = 0.05 and gives a p-value of 0.012. What is the decision?
Do not reject H0
Reject H0
✓
Accept H0 as proven true
Accept Ha as proven true
Correct Answer
Reject H0
Because alpha is greater than the p-value, the sample result is unusual enough under H0 to reject it. Do not reject H0 would be the decision only if the p-value were at least 0.05. Accept H0 as proven true is never a conclusion of a test, and here the evidence goes against H0. Accept Ha as proven true overstates the result, because a test gives evidence, not proof.
Category: P-Values and Decisions
Question 15
A test is run at alpha = 0.05 and gives a p-value of 0.20. Which conclusion is appropriate?
The data prove that H0 is true
There is sufficient evidence to support Ha
There is not sufficient evidence to support Ha
✓
The test was carried out incorrectly
Correct Answer
There is not sufficient evidence to support Ha
Since the p-value of 0.20 is larger than alpha, H0 is not rejected, which means the sample does not give enough evidence for the alternative. The data prove that H0 is true goes too far, because not rejecting H0 is not proof of it. There is sufficient evidence to support Ha would require a p-value below alpha. The test was carried out incorrectly has no basis, because large p-values are a normal outcome.
Category: P-Values and Decisions
Question 16
A test has Ha: mu not equal to 40. The area in the tail beyond the observed test statistic is 0.02. What is the p-value?
0.02
0.01
0.98
0.04
✓
Correct Answer
0.04
With a not-equal alternative, results equally extreme in either direction count, so the p-value is twice the one-tail area: 2 x 0.02 = 0.04. The value 0.02 is only the area in one tail. The value 0.01 halves the area instead of doubling it. The value 0.98 is the area that is not in that tail.
Category: P-Values and Decisions
Question 17
A right-tailed z test is run at alpha = 0.05, so the critical value is 1.645. Which value of the test statistic z would lead to rejecting H0?
1.80
✓
1.50
-1.80
0.95
Correct Answer
1.80
In a right-tailed test, H0 is rejected when the test statistic falls beyond the critical value in the right tail, and 1.80 is greater than 1.645. The value 1.50 is below the critical value. The value -1.80 lies in the left tail, which gives no evidence for a right-tailed alternative. The value 0.95 is well below the critical value.
Category: P-Values and Decisions
Question 18
A population has sigma = 8. A random sample of n = 16 has x-bar = 52. For H0: mu = 50, what is the z test statistic? Round to two decimal places.
0.25
1.00
✓
4.00
2.00
Correct Answer
1.00
The standard error is sigma / sqrt(n) = 8 / 4 = 2, so z = (52 - 50) / 2 = 1.00. The value 0.25 divides the difference by sigma alone, forgetting the sample size. The value 4.00 divides by sigma / n = 0.5 instead of sigma / sqrt(n). The value 2.00 is the raw difference x-bar - mu without dividing by the standard error.
Category: Test Statistic Calculations
Question 19
A random sample of n = 9 has x-bar = 23 and sample standard deviation s = 6. For H0: mu = 20, what is the t test statistic? Round to two decimal places.
0.50
4.50
1.50
✓
3.00
Correct Answer
1.50
The standard error is s / sqrt(n) = 6 / 3 = 2, so t = (23 - 20) / 2 = 1.50. The value 0.50 divides the difference by s alone. The value 4.50 divides by s / n instead of s / sqrt(n). The value 3.00 is the raw difference without dividing by the standard error.
Category: Test Statistic Calculations
Question 20
In a random sample of 100 people, 60 prefer a new product. For H0: p = 0.50, what is the z test statistic, using the standard error sqrt(p0(1 - p0) / n)? Round to two decimal places.
2.04
0.10
4.00
2.00
✓
Correct Answer
2.00
The sample proportion is p-hat = 0.60, and the standard error under H0 is sqrt(0.50 x 0.50 / 100) = 0.05, so z = (0.60 - 0.50) / 0.05 = 2.00. The value 2.04 comes from using p-hat instead of p0 in the standard error. The value 0.10 is the raw difference without dividing by the standard error. The value 4.00 comes from dividing 0.25 by sqrt(100) without taking the square root of the product.
Category: Test Statistic Calculations