In hypothesis testing, we have two hypotheses, a null hypothesis and an alternative hypothesis. The alternative hypothesis is what we are \"testing for\" (based on the research question). This is why we are collecting data to see if a certain population value differs from a given value (), is less than a given value (<), or is greater than a given value (>). The null hypothesis is what we are \"testing against\" which is the given value. 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 (p > 0.50). This is what we are \"testing for\" and collecting data for. The null hypothesis would be that the population proportion who voted for the candidate is 0.50 (p = 0.50) which would not be a majority. This is what we are \"testing against\". Note that the alternative hypothesis covers a range of values, but the null hypothesis is just the one value 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 .75. A medical research group invents a new treatment that they believe will be more successful, i.e., population proportion will exceed .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 .75 (p > . 75); Alternative: the population proportion of those successfully treated by the new treatment is .75 (p = .75) B) Null: the population proportion of those successfully treated by the new treatment is .75 (p = .75); Alternative: the population proportion of those successfully treated by the new treatment exceeds .75 (p > .75) C) Null: the population proportion of those successfully treated by the new treatment is .80 (p = .80); Alternative: the population proportion of those successfully treated by the new treatment exceeds .80 (p > .80) D) Null: the population proportion of those successfully treated by the new treatment exceeds .80 (p > . 80); Alternative: the population proportion of those successfully treated by the new treatment is .80 (p = .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 < population mean B); Alternative: the population mean number of hours of wearing comfort for group B is greater than that of group A (population mean B > 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 .35 since industry statistics indicate this proportion has been .35 for quite some time. A random sample of 1,000 car owners finds that the proportion that prefers blue cars is .40. What are the null and alternative hypotheses being tested? A) Null: the population proportion who prefer blue cars is .35; Alternative: the population proportion who prefer blue cars is greater than .35 B) Null: the population proportion who prefer blue cars is .40; Alternative: the population proportion who prefer blue cars differs from (does not equal) .40 C) Null: the population proportion who prefer blue cars is .40; Alternative: the population proportion who prefer blue cars is greater than .40 D) Null: the population proportion who prefer blue cars is .35; Alternative: the population proportion who prefer blue cars differs from (does not equal) .35 5. Previously a study found that Statistics students were spending about $300.00 on textbooks per semester. A researcher at Penn State 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 than we would expect given the spread of our distribution, then we have evidence to indicate that the null may not be the true population value. We calculate a test statistic value to \"standardize\" how far away from the null our observed sample value is. The test statistic (standardized score) provides a measure of how large the departure between our observed sample value and null value is. 6. Previously a study found that Statistics students were spending about $300.00 on textbooks per semester. A researcher at Penn State 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: A) $60 B) $4 C) $8 D) $15 7. Assume we have an observed sample mean of $325, a null value of $280, a SE of the mean under the null hypothesis of $15. What is our test statistic value (standardized score)? A) +6 B) +3 C) +45 D) -6 E) -3 F) -45 8. A polling organization surveyed Pennsylvanians regarding 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 under the null hypothesis? (we calculate the SE using the null value since that is hypothesized to be the true population value) 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 .40 , a null value of .35, (we are testing to see if the proportion of all adults who prefer blue cars has changed from .35 since industry statistics indicate the this proportion was .35 in the past), and an SE under the null hypothesis of 0.015 . What is our test statistic value (standardized score)? 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 (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 a test statistic value or value more 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 smaller value. For \"greater than\" alternative hypotheses we must only consider the probability of obtaining our test statistic value or larger value. 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 in two directions (both below our null value and above our null value). 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 < $100; the consultant conducts a study using a large random sample and calculates a test statistic of z = +0.5. The p-value for this test would be A) 0.84 B) 0.69 C) 0.16 D) 0.31 11. A consultant believes that the average amount spent by customers at an online book store will be more than the current $100 if they add a new faster shipping service. To test the null hypothesis that the population mean = $100, versus the alternative hypothesis that the population mean > $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 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: 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 these results or any results more extreme (in the direction of the alternative hypothesis) is HIGHLY UNLIKELY if we were to assume the null is true. Therefore, low p-values ( 0.05) indicate that the results found in the sample are highly unlikely if we were to assume the null is true, so we should reject the null and conclude the alternative. This does not technically indicate that the alternative is \"true\