Posted: May 17th, 2021

# In hypothesis testing, we have two hypotheses

In hypothesis testing, we have two hypotheses: a null hypothesis and an alternative hypothesis. The alternative hypothesis is typically what we want to demonstrate (based on the research question). We collect data to see if a certain population value differs from a given value (≠), is less than a given value ([removed]). The null hypothesis is typically a baseline or a known standard against which we are testing. For example: If we want to test to see if a majority of voters voted for a certain candidate, then our alternative hypothesis would be that the population proportion who voted for the candidate is greater than 0.50 (i.e. p > 0.50). This is what we want to demonstrate and is the reason for collecting data. The null hypothesis would be that the population proportion who voted for the candidate is 0.50 (i.e. p = 0.50) which would not be a majority. This is the baseline against which we are testing. Note that the alternative hypothesis covers a range of values, but the null hypothesis is just the one value (i.e. equality).

1. A polling group surveyed a city in Scotland regarding residents’ opinions on independence from the UK. It is generally believed that the percentage of ‘Yes’ votes is 50%. The poll wants to find out whether greater than half (> 50%) of the residents will vote ‘Yes.’ The survey polled 2000 residents, of which 1050 responded that they will vote ‘Yes’ on Scotland independence (52.5%). What are the null and alternative hypotheses?
A) Null: the percentage of ‘Yes’ votes is 52.5%; Alternative: the percentage of ‘Yes’ votes is greater than 52.5%
B) Null: the percentage of ‘Yes’ votes is greater than 52.5%; Alternative: the percentage of ‘Yes’ votes is 52.5%
C) Null: the percentage of ‘Yes’ votes is 50%; Alternative: the percentage of ‘Yes’ votes is greater than 50%
D) Null: the percentage of ‘Yes’ votes is greater than 50%; Alternative: the percentage of ‘Yes’ votes is 50%

2. For patients with a particular disease, the population proportion of those successfully treated with a standard treatment that has been used for many years is 0.75. A medical research group invents a new treatment that they believe will be more successful, i.e. the population proportion will exceed 0.75. A doctor plans a clinical trial he hopes will prove this claim. A sample of 100 patients with the disease is obtained. Each person is treated with the new treatment and eventually classified as having either been successfully or not successfully treated with the new treatment. Out of 100 patients, 80 (80%) were successfully treated by the new treatment. What are the null and alternative hypotheses?
A) Null: the population proportion of those successfully treated by the new treatment exceeds 0.75 (p > 0.75); Alternative: the population proportion of those successfully treated by the new treatment is 0.75 (p = 0.75)
B) Null: the population proportion of those successfully treated by the new treatment is 0.75 (p = 0.75); Alternative: the population proportion of those successfully treated by the new treatment exceeds 0.75 (p > 0.75)
C) Null: the population proportion of those successfully treated by the new treatment is 0.80 (p = 0.80); Alternative: the population proportion of those successfully treated by the new treatment exceeds 0.80 (p > 0.80)
D) Null: the population proportion of those successfully treated by the new treatment exceeds 0.80 (p > 0.80); Alternative: the population proportion of those successfully treated by the new treatment is 0.80 (p = 0.80)

3. Suppose that a study is done comparing two different contact lens wetting solutions with regard to hours of wearing comfort. 100 contact lens wearers are randomly divided into two groups. One group uses solution A for 2 months. The other group uses solution B for 2 months. The researcher wants to determine if there is a difference in the hours of wearing comfort for the two groups. The population mean number of hours of wearing comfort will be compared for the two groups. What are the null and alternative hypotheses being tested by the researcher?
A) Null: there is a difference in the population mean number of hours of wearing comfort for the two groups (two population means are not equal); Alternative: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal).
B) Null: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal); Alternative: the population mean from group A is larger than that of group B (population mean group A > population mean group B)
C) Null: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal); Alternative: there is a difference in the population mean number of hours of wearing comfort for the two groups (two population means are not equal)
D) Null: the population mean number of hours of wearing comfort for group A is less than that of group B (population mean A [removed] population mean A)

4. A car company is testing to see if the proportion of all adults who prefer blue cars has changed (differs) from 0.35 since industry statistics indicate this proportion has been 0.35 for quite some time. A random sample of 1,000 car owners finds that the proportion that prefers blue cars is 0.40. What are the null and alternative hypotheses being tested?
A) Null: the population proportion who prefer blue cars is 0.35; Alternative: the population proportion who prefer blue cars is greater than 0.35
B) Null: the population proportion who prefer blue cars is 0.40; Alternative: the population proportion who prefer blue cars differs from (does not equal) 0.40
C) Null: the population proportion who prefer blue cars is 0.40; Alternative: the population proportion who prefer blue cars is greater than 0.40
D) Null: the population proportion who prefer blue cars is 0.35; Alternative: the population proportion who prefer blue cars differs from (does not equal) 0.35

5. Previously a study found that statistics students were spending about \$300.00 per semester on textbooks. A researcher decides to research book cost because he now believes that book cost per semester is greater than \$300.00. A random sample of 225 statistics students finds that the sample average is \$324 with a standard deviation of \$60. In this situation, the null hypothesis is that:
A) the long run average is greater than \$300
B) the long run average is greater than \$324
C) the long run average equals \$300
D) the long run average equals \$324

In hypothesis testing, the null value is hypothesized to be our true population value. We know how sample statistic values will vary around this hypothesized true population value from Lesson 9. If the observed sample value is further away from the null value than we would expect by chance, given the spread of our distribution, then we have evidence to indicate that the null value may not be the true population value. We calculate a test statistic value to “standardize” how far away from the null value our observed sample value is. The test statistic (standardized score) provides a measure of the amount of difference between our observed sample value and null value.

6. Previously a study found that statistics students were spending about \$300.00 on textbooks per semester. A researcher decides to research book cost because he now believes that book cost per semester is greater than \$300.00. A random sample of 225 statistics students finds that the sample average is \$324 with a standard deviation of \$60. The standard error of the mean under the null hypothesis is which of the following? (Recall that the SEM = sample standard deviation/sqrt(sample size).)
A) \$60
B) \$4
C) \$8
D) \$15

7. Suppose we have an observed sample mean of \$325, a null value of \$280, and a standard error of the mean (SEM) under the null hypothesis of \$15. What is our test statistic value (standardized score)? (Test statistic = (sample mean – null value)/SEM)
A) +6
B) +3
C) +45
D) -6
E) -3
F) -45

8. A polling organization surveyed Ohio residents’ opinions on whether there should be term limits for State Representatives and Senators. The polling organization wants to find out whether greater than half (> 50%) of the residents are in favor of term limits. The pollsters randomly polled 500 residents, of which 275 responded that they favor term limits (55%). What approximately is the standard error of the proportion (SEP) under the null hypothesis? (We calculate the SEP using the null value since that is hypothesized to be the true population value. Recall that SEP = sqrt(((hypothesized proportion)*(1-hypothesized proportion))/sample size).)
A) 0.044 or 4.4%
B) 0.011 or 1.1%
C) 0.022 or 2.2%
D) 0.05 or 5%

9. Assume we have an observed sample proportion of 0.40 , a null value of 0.35 — we are testing to see if the proportion of all adults who prefer blue cars has changed from 0.35, since industry statistics indicate the this proportion was 0.35 in the past — and an SEP under the null hypothesis of 0.015. What is our test statistic value (standardized score)? (Recall that test statistic = (sample proportion – null value)/SEP)
A) +6.66
B) -6.66
C) -3.33
D) +3.33

We can then determine if a given magnitude of departure or any departure more extreme (i.e. even further away) from the null value is unusual (has low probability of occurring if the null is “true”) by determining the probabilities associated with test statistic values that correspond to our sample value and test statistic values that correspond to sample values that are even further away in the direction of the alternative hypothesis (remember, the alternative hypothesis is a range of values). A one-sided alternative hypothesis is when the alternative hypothesis states that a population proportion or population mean is either “greater than” or “less than” a set value or another group’s population proportion or population mean. For a one-sided test we must only consider the probability of obtaining the test statistic value or one even more extreme/unusual in the direction of the alternative hypothesis. Therefore, for “less than” alternative hypotheses, we must only consider the probability of obtaining our test statistic value or one even smaller. For “greater than” alternative hypotheses, we must only consider the probability of obtaining our test statistic or one even larger. A two-sided alternative hypothesis is when the alternative hypothesis states that a population proportion or population mean is “not equal to” (or differs from) a set value or another group’s population proportion or population mean. Two-sided tests require that we consider the probability of test statistic values that indicate a given magnitude of departure or more from the null value in both directions.

10. A consultant believes that the average amount spent by customers at an online shoe store will be less than the current \$100 if shipping costs are increased by 10% (the projected shipping cost increase proposed by their carrier). To test the null hypothesis that the population mean = \$100 versus the alternative hypothesis that the population mean [removed] \$100, the consultant conducts a study using a large random sample and calculates a test statistic of z = +1.96. The p-value for this test would be which of the following? (Recall: To find the probability of a z-score HIGHER than a certain value, you have to find the probability of getting a z-score LOWER than that value and then subtract that probability from 1. In other words, P(Z > 1.96) = 1 – P(Z < 1.96))
A) 0.975
B) 0.05
C) 0.025
D) 0.95

12. A cereal manufacturer tests their equipment weekly to be assured that the proper amount of cereal is in each box of cereal. The company wants to see if the amount differs from the stated amount on the box. The stated amount on each box for this particular cereal is 12.5 ounces. The manufacturer takes a random sample of 100 boxes and finds that they average 12.2 ounces with a standard deviation of 3 ounces. The standardized score is -1.00. The p-value for this test would be which of the following? (HINT: The p-value = P(Z [removed] 1). Since those two probabilities will be the same, you can simply find one of them and multiply it by 2!)
A) 0.16
B) 0.32
C) 0.30
D) 0.68

The p-value is then interpreted as: The probability (likelihood) of obtaining our test statistic value or any test statistic value more extreme (more unusual), if in fact the null hypothesis were true. Low p-values (≤ 0.05) indicate that obtaining the sample results (or any results even more extreme in the direction of the alternative hypothesis) is HIGHLY UNLIKELY if the null is true. Therefore, low p-values (≤ 0.05) lead us to reject the null hypothesis and conclude the alternative is true. This DOES NOT PROVE that the alternative is “true,” just that we have enough evidence to support the alternative and reject the null.

13. Suppose that the null hypothesis is, “The population mean is \$200,” and the alternative hypothesis is, “The population mean is less than \$200.” Also, suppose the test statistic value is -0.50, with a p-value of 0.31. The p-value is the probability of
A) obtaining our test statistic value or a value even smaller, if in fact the population mean is less than \$200
B) obtaining our test statistic value or a value even larger, if in fact the population mean is less than \$200
C) obtaining our test statistic value or a value even smaller, if in fact the population mean is \$200
D) obtaining our test statistic value or a value even larger, if in fact the population mean is \$200

14. Suppose that the null hypothesis is, “The population proportion is 0.50,” and the alternative hypothesis is, “The population proportion is greater than 0.50.” Further, suppose that our test statistic is +1.96, with a p-value of 0.025. The p-value is the probability of
A) obtaining our test statistic value or larger, if in fact the population proportion is 0.50
B) obtaining our test statistic value or smaller, if in fact the population proportion is 0.50
C) obtaining our test statistic value or larger, if in fact the population proportion is greater than 0.50
D) obtaining our test statistic value or smaller, if in fact the population proportion is greater than 0.50

15. A p-value of 0.05 or less is said to indicate that the results are “statistically significant.” What does statistically significant mean?
A) The null hypothesis is a poor explanation of the data.
B) The null hypothesis is a good explanation of the data.
C) The alternative hypothesis is a poor explanation of the data.

16. Three of the following statements about a p-value are true. Which one is false?
A) If we got a p-value of 0.52, we would not reject the null hypothesis.
B) If the p-value is very small, we reject the null hypothesis.
C) The p-value is the probability that the null hypothesis is true.
D) The p-value is the probability, assuming the null hypothesis is true, of seeing results as (or more) extreme as what we observed in the sample.

As sample size increases, we expect sample statistics to vary less around our hypothesized population value (the null value). Therefore, as sample size increases, a given magnitude of difference between our sample value and the null value becomes more unusual, resulting in a lower p-value. Therefore, a given magnitude of difference between the sample value and null value is more likely to achieve statistical significance (p-value ≤ 0.05) when the sample size is larger.

17. What does a decrease in sample size do to the probability of a Type 2 error?
A) Makes the probability of a Type 2 error lower
B) Makes the probability of a Type 2 error higher
C) Does not influence the probability of a Type 2 error

18. At the usual significance level of 5% (0.05), if a large number of significance tests are conducted on an equally large number of independent samples then we expect to a Type 1 error to occur (rejecting the null when it should not be rejected) in approximately _____ of the hypothesis tests.
A) 10%
B) 5%
C) Never
D) Impossible to determine

19. Once you reject the null and conclude the alternative hypothesis is true, is it possible to have a Type 2 error?
A) No
B) Yes

Previously a study found that statistics students were spending about \$400.00 on textbooks per semester. A researcher decides to research book cost because he now believes that book cost per semester is greater than \$400.00. A random sample of 64 statistics students finds that the sample mean is \$410.00 with a standard deviation of \$80. Is this strong evidence that the average amount spent on textbooks per semester is greater than \$400? We wish to carry out the appropriate hypothesis test.

1. In this situation, the null hypothesis is that:
A) the long run average is greater than \$400
B) the long run average is greater than \$410
C) the long run average equals \$400
D) the long run average equals \$410

2. The standard error of the mean (SEM) under the null hypothesis is:
A) \$80
B) \$10
C) \$20
D) \$15

3. What is our test statistic value (standardized score)?
A) +0.5
B) +1.0
C) -1.0
D) -0.5

4. What is the p-value for the significance test?
A) 0.31 or 31%
B) 0.84 or 84%
C) 0.16 or 16%
D) 0.69 or 69%

5. Based on the p-value, we can conclude:
A) that we cannot reject the null hypothesis
B) that the null hypothesis is a poor explanation of the data and can be rejected
C) that the null hypothesis is true
D) that the alternative hypothesis is true

6. What is the Type 2 error in this situation?
A) reject that the population average is \$400 when it really is equal to \$400
B) fail to reject that the population average is \$400 when it is actually greater than \$400

7. Suppose that the p-value for our test had been 0.21 (that was not the p-value, but let’s suppose it was). Based on a p-value of 0.21, would a 95% confidence interval for the true population average spent include \$400?
A) Yes
B) No

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