Solution

The correct answer is (E). Categorical variables are just another name for qualitative variables. And quantitative variables are numeric variables, so they can be continuous variables. Categorical variables, however, are not quantiative variables.

Solution

The correct answer is (D). If you toss a coin three times, there are a total of eight possible outcomes. They are: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT. Of the eight possible outcomes, three have exactly one head. They are: HTT, THT, and TTH. Therefore, the probability that three flips of a coin will produce exactly one head is 3/8 or 0.375.

Solution

The correct answer is (D). A simple random sample requires that every sample of size n (in this problem, n is equal to 400) have an equal chance of being selected. In this problem, there was a 100 percent chance that the sample would include 100 purchasers of each brand of car. There was zero percent chance that the sample would include, for example, 99 Ford buyers, 101 Honda buyers, 100 Toyota buyers, and 100 GM buyers. Thus, all possible samples of size 400 did not have an equal chance of being selected; so this cannot be a simple random sample.

The fact that each buyer in the sample was randomly sampled is a necessary condition for a simple random sample, but it is not sufficient. Similarly, the fact that each buyer in the sample had an equal chance of being selected is characteristic of a simple random sample, but it is not sufficient. The sampling method in this problem used random sampling and gave each buyer an equal chance of being selected; but the sampling method was actually stratified random sampling.

The fact that car buyers of every brand were equally represented in the sample is irrelevant to whether the sampling method was simple random sampling. Similarly, the fact that population consisted of buyers of different car brands is irrelevant.

Solution

The correct answer is (E). The confidence level is not affected by the margin of error. When the margin of error is small, the confidence level can low or high or anything in between. A confidence interval is a type of interval estimate, not a type of point estimate. A population mean is not an example of a point estimate; a sample mean is an example of a point estimate.

Solution

The correct answer is (B). First, we need to compute the sample mean.

x = ( 1 + 3 + 5 + 7 ) / 4 = 4

Then, we plug all of the known values into the formula for the standard deviation of a sample, as shown below:

s = sqrt [ Σ ( x_i - x )² / ( n - 1 ) ]
s = sqrt { [ ( 1 - 4 )² + ( 3 - 4 )² + ( 5 - 4 )² + ( 7 - 4 )² ] / ( 4 - 1 ) }
s = sqrt { [ ( -3 )² + ( -1 )² + ( 1 )² + ( 3 )² ] / 3 }
s = sqrt { [ 9 + 1 + 1 + 9 ] / 3 } = sqrt (20 / 3) = sqrt ( 6.67 ) = 2.58

Solution

The correct answer is C. Let S = the event that the card is a spade; and let A = the event that the card is an ace. We know the following:

• There are 52 cards in the deck.
• There are 13 spades, so P(S) = 13/52.
• There are 4 aces, so P(A) = 4/52.
• There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.

Therefore, based on the rule of addition:

Solution

The correct answer is (A). The standard error can be computed from a knowledge of sample attributes - sample size and sample statistics. The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. The standard error is a measure of variability, not a measure of central tendency.

Solution

The correct answer is (B). To compute the margin of error, we need to find the critical value and the standard error of the mean. To find the critical value, we take the following steps.

• Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 0.95 = 0.05
• Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
• Find the critical z score. Since the sample size is large, the sampling distribution will be roughly normal in shape. Therefore, we can express the critical value as a z score. For this problem, it will be the z score having a cumulative probability equal to 0.975. Then, using an online calculator (e.g., Stat Trek's free normal distribution calculator), a handheld graphing calculator, or the standard normal distribution table, we find the cumulative probability associated with the z-score. Using the Normal Distribution Calculator, we find that the critical value is 1.96. (On the actual AP Statistics exam, you may need to use a graphing calculator or a normal table, since Stat Trek's analytical tools will not be available.)

Next, we find the standard error of the mean, using the following equation:

SE_x = s / sqrt( n ) = 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

And finally, we compute the margin of error (ME).

ME = Critical value x Standard error = 1.96 * 0.013 = 0.025

Solution

The correct answer is (E). From the z-score equation, we know

z = (X - μ) / σ

where z is the z-score, X is the value of the element, μ is the mean of the population, and σ is the standard deviation.

Solving for Jane's test score (X), we get

X = ( z * σ) + 100 = ( 1.20 * 15) + 100 = 18 + 100 = 118

Solution

The correct answer is B. The annual number of sweepstakes winners is an integer value and it results from a random process; so it is a discrete random variable. The average height of a group of boys could be a non-integer, so it is not a discrete variable. And the number of presidential elections in the 20th century is an integer, but it does not vary and it does not result from a random process; so it is not a random variable.

Solution

The correct answer is (E). In a sample survey, the researcher does not assign treatments to survey respondents. Therefore, a sample survey is not an experimental study; rather, it is an observational study. An observational study may or may not require fewer resources (time, money, manpower) than an experiment. The best method for investigating causal relationships is an experiment - not an observational study - because an experiment features randomized assignment of subjects to treatment groups. Randomization "evens out" the effects of extraneous variables, which makes it easier for the researcher to identify causal effects of treatment variables.

Solution

The answer is (A). To specify the confidence interval, we work through the four steps below.

• Identify a sample statistic. Since we are trying to estimate the mean weight in the population, we choose the mean weight in our sample (180) as the sample statistic.
  
  
• Select a confidence level. In this case, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
  
  
• Find the margin of error. Previously, we described how to compute the margin of error. The key steps are shown below.
  
  
  • Find standard error. The standard error (SE) of the mean is:

    SE = s / sqrt( n ) = 30 / sqrt(1000) = 30/31.62 = 0.95

  • Find critical value. The critical value is a factor used to compute the margin of error. To express the critical value as a t score (t*), follow these steps.
    
    
    • Compute alpha (α): α = 1 - (confidence level / 100) = 0.05
    • Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
    • Find the degrees of freedom (df): df = n - 1 = 1000 - 1 = 999
    • The critical value is the t score having 999 degrees of freedom and a cumulative probability equal to 0.975. Using an online calculator (e.g., Stat Trek's free t Distribution Calculator), a handheld graphing calculator, or a t distribution table, we find that the t score associated with a cumulative probability of 0.975 is 1.96. (On the actual AP Statistics exam, you may need to use a graphing calculator or a t table, since Stat Trek's analytical tools will not be available.)

    Note: We might also have expressed the critical value as a z score. Because the sample size is large, a z score analysis produces the same result - a critical value equal to 1.96.

  • Compute margin of error (ME): ME = critical value * standard error = 1.96 * 0.95 = 1.86
  
  
• Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level. Therefore, we can be 95% confident that the population means falls within the interval 180 + 1.86.

Solution

The correct answer is (D). The range is equal to the biggest value minus the smallest value. The biggest value is 81, and the smallest value is 11; so the range is equal to 81 -11 or 70. The median is equal to the middle value in the data set. Here, we have an even number of values - 45 and 47 - in the middle of the data set. Their average is (45 + 47)/2 or 46, so the median is equal to 46. The mean is equal to the average of all the values - 45.56.

Solution

The correct answer is A. The sum of all the probabilities is equal to 1. Therefore, the probability that four or more adults reside in a home is equal to 1 - (0.25 + 0.50 + 0.15) or 0.10.

Solution

The answer is (D). The approach that we used to solve this problem is valid when the following conditions are met.

• The sampling method must be simple random sampling. This condition is satisfied; the problem statement says that we used simple random sampling.
• The sample should include at least 10 successes and 10 failures. Suppose we classify a "more local news" response as a success, and any other response as a failure. Then, we have 0.40 * 1600 = 640 successes, and 0.60 * 1600 = 960 failures - plenty of successes and failures.
• If the population size is much larger than the sample size, we can use an "approximate" formula for the standard deviation or the standard error. This condition is satisfied, so we will use a simple "approximate" formula for the standard error.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval.

• Identify a sample statistic. Since we are trying to estimate a population proportion, we choose the sample proportion (0.40) as the sample statistic.
  
  
• Select a confidence level. The confidence level is defined for us in the problem statement. We are working with a 99% confidence level.
  
  
• Find the margin of error. Elsewhere on this site, we show how to compute the margin of error when the sampling distribution is approximately normal. The key steps are shown below.
  
  
  • Find standard deviation or standard error. Since we do not know the population proportion, we cannot compute the standard deviation; instead, we compute the standard error. And since the population is more than 10 times larger than the sample, we can use the following formula to compute the standard error (SE) of the proportion:

    SE = sqrt [ p(1 - p) / n ] = sqrt [ (0.4)*(0.6) / 1600 ] = sqrt [ 0.24/1600 ] = 0.012

  • Find critical value. The critical value is a factor used to compute the margin of error. Because the sampling distribution is approximately normal and the sample size is large, we can express the critical value as a z score by following these steps.
    
    
    • Compute alpha (α): α = 1 - (confidence level / 100) = 1 - (99/100) = 0.01
    • Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.01/2 = 0.995
    • The critical value is the z score having a cumulative probability equal to 0.995. Using an online calculator (e.g., Stat Trek's free Normal Distribution Calculator), a handheld graphing calculator, or a normal distribution table, we find that the z score associated with a cumulative probability of 0.995 is 2.58. (On the actual AP Statistics exam, you may need to use a graphing calculator or a normal distribution table, since Stat Trek's analytical tools will not be available.)
    
    
  • Compute margin of error (ME): ME = critical value * standard error = 2.58 * 0.012 = 0.03
  
  
• Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.

Therefore, the 99% confidence interval is 0.37 to 0.43. That is, we are 99% confident that the true population proportion is in the range defined by 0.04 + 0.03.

Solution

The correct answer is (C). The approach that we used to solve this problem is valid when the following conditions are met.

• The sampling method must be simple random sampling. This condition is satisfied; the problem statement says that we used simple random sampling.
• The sampling distribution should be approximately normally distributed. Because the sample size is large, we know from the central limit theorem that the sampling distribution of the mean will be normal or nearly normal; so this condition is satisfied.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval.

• Identify a sample statistic. Since we are trying to estimate a population mean, we choose the sample mean (115) as the sample statistic.
  
  
• Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 99% confidence level.
  
  
• Find the margin of error. Elsewhere on this site, we show how to compute the margin of error when the sampling distribution is approximately normal. The key steps are shown below.
  
  
  • Find standard deviation or standard error. Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean; instead, we compute the standard error (SE). Because the sample size is much smaller than the population size, we can use the "approximate" formula for the standard error.

    SE = s / sqrt( n ) = 10 / sqrt(150) = 10 / 12.25 = 0.82

  • Find critical value. The critical value is a factor used to compute the margin of error. Because the standard deviation of the population is unknown, we express the critical value as a t score rather than a z score. To find the critical value, we take these steps.
    
    
    • Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 99/100 = 0.01
    • Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.01/2 = 0.995
    • Find the degrees of freedom (df): df = n - 1 = 150 - 1 = 149
    • The critical value is the t score having 149 degrees of freedom and a cumulative probability equal to 0.995. Using an online calculator (e.g., Stat Trek's free t Distribution Calculator), a handheld graphing calculator, or a t distribution table, we find that the t score associated with a cumulative probability of 0.995 is 2.61. (On the actual AP Statistics exam, you may need to use a graphing calculator or a t table, since Stat Trek's analytical tools will not be available.)

    Note: We might also have expressed the critical value as a z score. Because the sample size is fairly large, a z score analysis produces a similar result - a critical value equal to 2.58.

  • Compute margin of error (ME): ME = critical value * standard error = 2.61 * 0.82 = 2.1
  
  
• Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.

Therefore, the 99% confidence interval is 112.9 to 117.1. That is, we are 99% confident that the true population mean is in the range defined by 115 + 2.1.