Statistic Question

Which of the following best describes a sampling distribution of a statistic?

• It is the probability that the sample statistic equals the parameter of interest.

• It is the probability distribution of all the values that are contained in all possible samples of the same size.

• It is the distribution of all of the statistics calculated from all possible samples of the same size.

• It is the histogram of sample statistics from all possible samples of the same size.

• All answers are correct.

A simple random sample of 50 adults were asked to reveal their gross annual incomes. The variance of this sample:

• is always smaller than the variance of the population.

• cannot be computed since the population size is not given.

• equals the variance of the population.

• is an estimate of the variance in the sampling distribution of the means of the gross annual incomes of all possible samples.

• is an estimate of the variance of the population but may differ from the variance of the population.

A simple random sample of 100 high school seniors in a certain suburb reveals that 65% of them have at least part-time jobs in addition to school. If the expected value of this proportion is equal to the proportion of high school seniors who have at least part-time jobs for the entire suburb, then we say that the sample proportion is:

• a true value.

• an unbiased estimator of the population proportion.

• equal to the population proportion.

• an estimate whose variance equals the variance of data in the population.

• less than the population proportion since only 100 students were sampled.

Suppose you roll a die 10 times and record the proportion of sixes. Suppose you then conduct a simulation of this experiment, first 100 times, then 1,000 times, and draw one histogram of the proportion of sixes found after 100 simulations and a second histogram of the proportions of sixes found after 1,000 simulations. Which of the following is true regarding the mean of the proportions of sixes from each simulation?

• The mean of the proportion of sixes for the 100 simulations will equal the mean of the proportion of sixes for the 1,000 simulations.

• The mean of the proportion of sixes for the 1,000 simulations will be a better estimator of the theoretical probability of rolling a six than the mean of the proportion of sixes for the 100 simulations.

• The mean of the proportion of sixes for the 100 simulations will be less than the mean of the proportion of sixes for the 1,000 simulations.

• The mean of the proportions of sixes for both simulations will not estimate the theoretical probability of rolling a six since they are finite samples from an infinite population.

• None of the answers is true.

Suppose you roll a die 10 times and record the proportion of sixes. Suppose you then conduct a simulation of this experiment, first 100 times, then 1,000 times, and draw one histogram of the proportion of sixes found after 100 simulations and a second histogram of the proportions of sixes found after 1,000 simulations. Which of the following statements are true regarding the histograms of the results from the two simulations?

I. The histograms from both simulations will be skewed left since a fair die does not exist in nature.

II. The histograms from both simulations will be mound-shaped and symmetric.

III. The histogram from the experiment that has 1,000 simulations will tend to be more mound-shaped and symmetric than the histogram from the experiment that has 100 simulations.

• I only

• I and II

• II and III

• III only

• None of these statements is true.

Two SRS of 50 undergraduates each from two universities are taken to determine the proportion of students who approve of the food service at their respective schools. The first university has an enrollment of 5,000 undergraduates while the second university has an enrollment of 35,000 undergraduates. Which of the following is the most accurate statement regarding theses samples?

• The variability of the sample from the larger university will be greater than the variability of the sample from the smaller university.

• The proportion of students who approve of the food service will be the same since the sample sizes are the same.

• The enrollment figures from the two universities are not relevant to whether the sample statistics obtained are unbiased estimates of the parameters of the two populations.

• If a university with 100,000 undergraduates conducted a SRS of 50 of its students, the results would be less accurate than either sample referenced above.

• None of these is an accurate statement.

Which of the following is true regarding the variation of a sampling distribution of a sample proportion?

• Variation depends on population size as well as sample size.

• The variance of a sampling distribution of a sample proportion for all samples of size 1 is 0.

• As the size of the sample increases, the variation of the sampling distribution approaches the variation of the population.

• For a given sample size, the maximum variation in the sampling distribution of a sample proportion occurs when the sample proportion is .5.

• None of these is true.

The conditions that np > 10 and n(1 - p) > 10 are imposed on a sampling distribution to protect against

• a sample that is not representative of the population.

• bias in the responses of the sample participants.

• skewness in the distribution.

• a very small population size.

• The conditions are not designed to protect against any of these conditions.

Records at a large university indicated that 20% of all freshmen are placed on academic probation at the end of their first semester. A random sample of 100 of this year's freshmen indicated that 25% of them were placed on academic probation at the end of the first semester. the results of this sample:

• are surprising since it indicates the 5% more of these freshmen were placed on academic probation than was expected.

• are surprising since SAT scores have been increasing over the past few years.

• are not surprising since the standard deviation of the sampling distribution is 4%.

• are surprising since the standard deviation of the sampling distribution is 0.4%

• are biased since the increase of 5% could not happen without injecting bias into the sample.

An investigato0.07ticipates that the proportion of red blossoms in his hybrid plants is 0.15. A random sample of 50 of his plants indicated that 22% of the blossoms were red. The standard deviation of the sampling distribution of the sample proportion is approximately:

• 0.051

• 0.059

• 0.07

• 0.116

• Cannot be determined.