AP Review IV - Hypothesis Tests/Confidence Intervals (30% – 40%)
I. Why do we do statistics? To answer a burning question. So the first step is to decide what question we’re trying to answer. If we don’t have a burning question, then we can shut our book, turn off our calculator, and go home. Now the only way to really know the truth is to study the entire population with which we’re concerned. But hey, we don’t really care that much….plus, we’ve got a life that has a time limit. So studying a sample to answer our question works for us.
II. Confidence Interval or Hypothesis Test?
· Confidence Interval – When you don’t know the population characteristic you want and you wish you did, this is your choice. Used to estimate the population parameter. The CI is an interval of plausible values that we hope will capture the value of the population characteristic. Helps us to get an idea of what we want to know.
· Hypothesis Test – When you have a set standard and want to know if your sample data meets that standard (or not), this is your choice. Helps to settle arguments of the type “yes, it does”, “no it doesn’t”, “yes, it does”, “no it doesn’t”. You should expect a bit of a difference, but a hypothesis test will help you decide if the difference is significant.
III. Questions to ask:
1. Will the data be categorical or quantitative?
2. How many samples are we dealing with? How many variables?
3. What statistics do we need to carry out our test or make a confidence interval?
4. What procedures must we follow so our answer will be credible?
5. Most important: What question are you trying to answer?
TO RECEIVE FULL CREDIT FOR A HYPOTHESIS TEST YOU MUST:
1) write the null and alternative hypothesis and define each variable
2) write which test you are using in words or with the appropriate formula and why you chose that test
3) write and check all conditions for that test
4) give the test statistic and the p-value (or critical value if you prefer) and df , if applicable
5) reject or fail to reject Ho based on the p-value (or critical value)
6) write a conclusion in terms of the problem
You either have enough evidence to claim whatever the alternative hypothesis represents (reject Ho)or you do not have enough evidence to claim whatever the alternative hypothesis represents (fail to reject Ho)
TO RECEIVE FULL CREDIT FOR A CONFIDENCE INTERVAL YOU MUST:
1) correctly identify the type of interval by name or formula
2) write and check all conditions for that interval
3) correctly calculate the interval – show work
4) correctly interpret the interval in terms of the problem
5) you may also be required to correctly interpret the confidence level
NOTE: All confidence intervals have the same assumptions of the corresponding hypothesis test.
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Type I errors, Type II errors, and Power
Type I error – () – when Ho is true but you go with Ha.
Type II error – (β) – when the alternative, Ha, is true but you go with Ho
Type I and Type II errors are inversely related; as one increases the other decreases.
Power = 1 – β, so Type II errors and power are inversely related. Type I errors and power are directly related.
For each of the following scenarios, determine the type of inference procedure to use. Then, proceed with the inference procedure. (will need to do this on another page)
1. (1998 #5) A large university provides housing for 10 percent of its graduate students to live on campus. The university's housing office thinks that the percentage of graduate students looking for housing on campus may be more than 10 percent. The housing office decides to survey a random sample of graduate students, and 62 of the 481 respondents say that they are looking for housing on campus.
On the basis of the survey data, would you recommend that the housing office consider increasing the amount of housing on campus available to graduate students? Give appropriate evidence to support your recommendation.
2. (2007 #4) Investigators at the U.S. Department of Agriculture wished to compare methods of determining the level of E. coli bacteria contamination in beef. Two different methods (A and B) of determining the level of contamination were used on each of ten randomly selected specimens of a certain type of beef. The data obtained, in millimicrobes/liter of ground beef, for each of the methods are shown in the table below.
Is there a significant difference in the mean amount of E. coli bacteria detected by the two methods for this type of beef? Provide a statistical justification to support your answer.
3. (2009B #5) A bottle-filling machine is set to dispense 12.1 fluid ounces into juice bottles. To ensure that the machine is filling accurately, every hour a worker randomly selects four bottles filled by the machine during the past hour and measures the contents. If there is convincing evidence that the standard deviation is greater than 0.05 ounce, the machine is shut down for recalibration. It can be assumed that the amount of juice that is dispensed into bottles is normally distributed.
During one hour, the mean number of fluid ounces of four randomly selected bottles was 12.05 and the standard deviation was 0.085 ounce.
a) Perform a test of significance to determine whether the mean amount of juice dispensed is different from 12.1 fluid ounces. Assume the conditions for inference are met.
4. (2010 #5) A large pet store buys the identical species of adult tropical fish from two different suppliers: Buy-Rite Pets and Fish Friends. Several of the managers at the pet store suspect that the lengths of the fish from Fish Friends are consistently greater than the lengths of the fish from Buy-Rite Pets. Random samples of 8 adult fish of the species from Buy-Rite Pets and 10 adult fish of the same species from fish Friends were selected and the lengths of the fish, in inches, were recorded, as shown in the table below.
Length of fish / Mean / StandardDeviation
Buy-Rite Pets
n = 8 / 3.4 2.7 3.3 4.1 3.5 3.4 3.0 3.8 / 3.40 / 0.434
Fish Friends
n = 10 / 3.3 2.9 4.2 3.1 4.2 4.0 3.4 3.2 3.7 2.6 / 3.46 / 0.550
Do the data provide convincing evidence that the mean length of the adult fish of the species from Fish Friends is greater than the mean length of the adult fish of the same species from Buy-Rite Pets?
5. (2008B #3 modified) A car manufacturer is interested in conducting a study to estimate the mean stopping distance for a new type of brakes when used in a car that is traveling at 60 miles per hour. These new brakes will be installed on cars of the same model and the stopping distance will be observed. The cost of each observation is $100. A budget of $12,000 is available to conduct the study and the goal is to carry it out in the most economical way possible. Preliminary studies indicate the σ = 12 feet for stopping distances.
A regulatory agency requires a 95% level of confidence for an estimate of mean stopping distance that is within 2 feet of the true mean stopping distance. The car manufacturer cannot exceed the budget of $12,000 for the study. Discuss the difficulties.
6. (2005B #4) A researcher believes that treating seeds with certain additives before planting can enhance the growth of plants. An experiment to investigate this is conducted in a greenhouse. From a large number of Roma tomato seeds, 24 seeds are randomly chosen and 2 are assigned to each of 12 containers. One of the 2 seeds is randomly selected and treated with the additive. The other seed serves as a control. Both seeds are then planted in the same container. The growth, in centimeters, of each of the 24 plants is measured after 30 days. These data were used to generate the partial computer output shown below. Graphical displays indicate that the assumption of normality is not unreasonable.
(a) Construct a confidence interval for the mean difference in growth, in centimeters, of the plants from the untreated and treated seeds. Be sure to interpret this interval.
7. 1999 #2) The Colorado Rocky Mountain Rescue Service wishes to study the behavior of lost hikers. If more were known about the direction in which lost hikers tend to walk, then more effective search strategies could be devised. Two hundred hikers selected at random from those applying for hiking permits are asked whether they would head uphill, downhill, or remain in the same place if they became lost while hiking. Each hiker in the sample was also classified according to whether he or she was an experienced or novice hiker. The resulting data are summarized in the following table.
Do these data provide convincing evidence of an association between the level of hiking expertise and the direction the hiker would head if lost?
Give appropriate statistical evidence to support your conclusion.
8. (2006 #4) Patients with heart-attack symptoms arrive at an emergency room either by ambulance or self-transportation provided by themselves, family, or friends. When a patient arrives at the emergency room, the time of arrival is recorded. The time when the patient's diagnostic treatment begins is also recorded.
An administrator of a large hospital wanted to determine whether the mean wait time (time between arrival and diagnostic treatment) for patients with heart-attack symptoms differs according to the mode of transportation. A random sample of 150 patients with heart-attack symptoms who had reported to the emergency room was selected. For each patient, the mode of transportation and wait time were recorded. Summary statistics for each mode of transportation are shown in the table below.
(a) Use a 99 percent confidence interval to estimate the difference between the mean wait times for ambulance-transported patients and self-transported patients at this emergency room.
9. (2005B #5) John believes that as he increases his walking speed, his pulse rate will increase. He wants to model this relationship. John records his pulse rate, in beats per minute (bpm), while walking at each of seven different speeds, in miles per hour (mph). A scatterplot and regression output are shown below.
(b) Do your estimates of the slope and intercept parameters have meaningful interpretations in the context of this question? If so, provide interpretations in this context. If not, explain why not.
10. (1997 #4) A random sample of 415 potential voters was interviewed 3 weeks before the start of a state-wide campaign for governor; 223 of the 415 said they favored the new candidate over the incumbent. However, the new candidate made several unfortunate remarks one week before the election. Subsequently, a new random sample of 630 potential voters showed 317 voters favored the new candidate.
Do these data support the conclusion that there was a decrease in voter support for the new candidate after the unfortunate remarks were made? Give appropriate statistical evidence to support your answer.