Clarifying the Concepts
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Calculating the Statistic
Calculate the following percentages for a z score of 0.74, with a tail of 22.96%:
Using the z table in Appendix B, calculate the following percentages for a z score of −0.08:
Using the z table in Appendix B, calculate the following percentages for a z score of 1.71:
Rewrite each of the following percentages as probabilities, or p levels:
Rewrite each of the following probabilities, or p levels, as percentages:
For each of the following p levels, what percentage of the data will be in each critical region for a two-
State the percentage of scores in a one-
If the cutoffs for a z test are −1.96 and 1.96, determine whether you would reject or fail to reject the null hypothesis in each of the following cases:
If the cutoffs for a z test are −2.58 and 2.58, determine whether you would reject or fail to reject the null hypothesis in each of the following cases:
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Use the cutoffs of −1.65 and 1.65 and a p level of approximately 0.10, or 10%. For each of the following values, determine whether you would reject or fail to reject the null hypothesis:
You are conducting a z test on a sample for which you observe a mean weight of 150 pounds. The population mean is 160, and the standard deviation is 100.
For each of the following, indicate whether or not the situation describes misleading data that the researcher may decide to investigate and potentially discard.
Assume that the following set of data represents the responses of 10 participants to three similar statements. The participants rated their agreement with each statement on a scale from 1 to 7.
| Participant | S1 | S2 | S3 |
|---|---|---|---|
| 1 | 2 | 3 | 2 |
| 2 | 6 | 7 | 3 |
| 3 | 3 | 2 | 5 |
| 4 | 7 | 6 | 5 |
| 5 | 2 | 3 | 3 |
| 6 | 5 | 5 | 6 |
| 7 | 9 | 5 | 4 |
| 8 | 2 | 3 | 7 |
| 9 | 6 | 7 | 7 |
| 10 | 3 | 6 | 5 |
Applying the Concepts
Percentiles and unemployment rates: The U.S. Bureau of Labor Statistics’ annual report published in 2011 provided adjusted unemployment rates for 10 countries. The mean was 7, and the standard deviation was 1.85. For the following calculations, treat these as the population mean and standard deviation.
Height and the z distribution: Elena, a 15-
Height and the z distribution: Kona, a 15-
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Height and the z statistic: Imagine a class of thirty-
Height and the z statistic: Imagine a basketball team comprised of thirteen 15-
The z distribution and statistics test scores: Imagine that your statistics professor lost all records of students’ raw scores on a recent test. However, she did record students’ z scores for the test, as well as the class average of 41 out of 50 points and the standard deviation of 3 points (treat these as population parameters). She informs you that your z score was 1.10.
The z statistic, distributions of means, and height: Using what we know about the height of 15-
The z statistic, distributions of means, and height: Another teacher decides to average the height of all 15-
Directional versus nondirectional hypotheses: For each of the following examples, identify whether the research has expressed a directional or a nondirectional hypothesis:
Null hypotheses and research hypotheses: For each of the following examples (the same as those in Exercise 7.45), state the null hypothesis and the research hypothesis, in both words and symbolic notation:
The z distribution and Hurricane Katrina: Hurricane Katrina hit New Orleans on August 29, 2005. The National Weather Service Forecast Office maintains online archives of climate data for all U.S. cities and areas. These archives allow us to find out, for example, how the rainfall in New Orleans that August compared to that in the other months of 2005. The table below shows the National Weather Service data (rainfall in inches) for New Orleans in 2005.
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| January | 4.41 |
| February | 8.24 |
| March | 4.69 |
| April | 3.31 |
| May | 4.07 |
| June | 2.52 |
| July | 10.65 |
| August | 3.77 |
| September | 4.07 |
| October | 0.04 |
| November | 0.75 |
| December | 3.32 |
Percentiles and IQ scores: IQ scores are designed to have a mean of 100 and a standard deviation of 15. IQ testing is one way in which people are categorized as having different levels of mental disability; there are four levels of mental retardation between the IQ scores of 0 and 70.
Step 1 of hypothesis testing for a study of the Wechsler Adult Intelligence Scale: Boone (1992) examined scores on the Wechsler Adult Intelligence Scale-
Step 2 of hypothesis testing for a study of the Wechsler Adult Intelligence Scale: Refer to the scenario described in Exercise 7.49.
Step 1 of hypothesis testing for a study of college football: Let’s consider whether U.S. college football teams are more likely or less likely to be mismatched in the upper National Collegiate Athletic Association (NCAA) divisions. Overall, the 53 Football Bowl Subdivision (FBS) games (formerly Division I-
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Step 2 of hypothesis testing for a study of college football: Refer to Exercise 7.51.
Steps 3 through 6 of hypothesis testing for a study of college football: Refer to Exercise 7.51. Remember, the population mean is 16.189, with a standard deviation of 12.128. The results for the four FCS Patriot League games are as follows:
Holy Cross, 27/Bucknell, 10
Lehigh, 23/Colgate, 15
Lafayette, 31/Fordham, 24
Georgetown, 24/Marist, 21
Putting It All Together
The Graded Naming Test and sociocultural differences: Researchers often use z tests to compare their samples to known population norms. The Graded Naming Test (GNT) asks respondents to name objects in a set of 30 black-
Patient adherence and orthodontics: A research report (Behenam & Pooya, 2007) begins, “There is probably no other area of health care that requires…cooperation to the extent that orthodontics does,” and explores factors that affected the number of hours per day that Iranian patients wore their orthodontic appliances. The patients in the study reported that they used their appliances, on average, 14.78 hours per day, with a standard deviation of 5.31. We’ll treat this group as the population for the purposes of this example. Let’s say a researcher wanted to study whether a DVD with information about orthodontics led to an increase in the amount of time patients wore their appliances, but decided to use a two-
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Radiation levels on Japanese farms: Fackler (2012) reported in the New York Times that Japanese farmers have become skeptical of the Japanese government’s assurances that radiation levels were within legal limits in the wake of the 2011 tsunami and radiation disaster at Fukushima. After reports of safe levels in Onami, more than 12 concerned farmers tested their crops and found dangerously high levels of cesium.
You have conducted a study with 120 participants (60 female, 60 male) about the relation between attitudes toward cohabitation before marriage (on a 30-
You have just conducted a study testing how well two independent variables, daily sugar intake (as assessed by a 25-
In Next Steps, we noted that the z distribution is sometimes used to identify potential outliers in a data set. Box Office Mojo (2013) provides data on U.S. box office receipts for major films. Here are worldwide box office grosses for a randomly selected sample of 15 of the 100 top-
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| Movie | Millions of dollars |
|---|---|
| Marvel’s The Avengers | 1512 |
| Flight | 162 |
| Skyfall | 1109 |
| Wrath of the Titans | 305 |
| Tyler Perry’s Madea’s Witness Protection | 66 |
| Zero Dark Thirty | 109 |
| Lincoln | 275 |
| Moonrise Kingdom | 68 |
| Life of Pi | 609 |
| The Lucky One | 99 |
| The Bourne Legacy | 276 |
| The Watch | 68 |
| Rock of Ages | 59 |
| Cloud Atlas | 131 |
| Snow White and the Huntsman | 397 |