Statistics
Two events, A and B, are independent. The probability of A is 0.2, the probability of B is 0.3. What is the probability of A and B?
0.06
The local general store is interested in finding how many people prefer buying chips with a cold drink. They record data for a week. Let x denote the number of cold drinks sold on a given day and let y denote the number of chips sold on that same day. The correlation between the 2 variables is found to be 0.470. What will the new correlation be in the next week if the values of x and y are interchanged?
0.470 When we interchange the values of x with the values of y, there would be no change in the above formula, except the position of the expressions. Hence, the new correlation would be the same.
Two fair dice are rolled with the results hidden from you. You are shown that one of the dice is a 3. What is the probability that the hidden die is also a 3?
1/6 Explanation 1 in 6. The dice are independent, so the fact that one of the dice (the revealed one) is a 3 has no effect on the probability of the other dice being a 3. Of note, 1 in 36 is the probability of both dice being 3s. Since we are shown the revealed die is a three (i.e., probability 1), however, the conditional probability of the hidden die being a 3 becomes 1 in 6.
Two events, A and B, are mutually exclusive. The probability of event A is 1/7. The probability of event B is 1/5. What is the probability of event A or event B?
12/35
What is the standard deviation calculated for the following 10 fasting blood sugar levels of patients with diabetes? 84, 87, 93, 99, 103, 116, 126, 129, 130, 133
18.02 Explanation A standard deviation is the square root of a variance, so the variance must be calculated first. Variance is the squared difference from the mean (average). First, calculate the mean of the values. 84 + 87 + 93 + 99 + 103 + 116 + 126 + 129 + 130 + 133 = 1100 1100 ÷ 10 = 110 For each value, subtract the mean and square the result (the squared difference). For example, 84 - 110 = -26 -262 = 676 After completing all the calculations, take the average of the squared difference to find the variance. 3246/10 = 324.6 Therefore, the variance is 324.6. The standard deviation (the square root of the variance) is √324.6=18.02
What is the variance calculated for the following 10 fasting blood sugar levels of patients with diabetes? 84, 87, 93, 99, 103, 116, 126, 129, 130, 133
324.6 Explanation Variance is the squared difference from the mean (average). First, calculate the mean of the values. 84 + 87 + 93 + 99 + 103 + 116 + 126 + 129 + 130 + 133 = 1100 1100 ÷ 10 = 110 For each value, subtract the mean and square the result (the squared difference). For example, 84 - 110 = -26 -262 = 676 After completing all the calculations, take the average of the squared difference. The answer is 324.6.
What are the first quartile, second quartile, and third quartile of the following data set? 3,5,5,7,9,12,14
5, 7, 12
A fair die is rolled. What is the probability that the first roll is not a 6 and the second roll is a 6?
5/36
A machine produces items. Each item is independently produced and has a 5% chance of failing quality control checks. If 1000 items are produced during a shift, what is the expected number of good items?
950
A woman has contracted a contagious disease that spreads by direct physical contact. After her condition became contagious, she was in direct physical contact with 12 people. For any individual contact, the disease has a 6% probability of being transmitted. Question What is the probability that this woman has infected exactly 2 other people?
About 13%
A researcher wants to study the effects of a new drug on people with heart disease. They acquire a random sample of local volunteers with heart disease using a random number generator. For the first trial study, the researcher randomly assigns each participant to the control group or the treatment group—by a coin toss—and gives each a placebo or the trail drug, respectively, then monitors and evaluates the results.
All local people with heart disease
An instructor was concerned that the highest score on an exam was 95, rather than 100, so they decided to add 5 points to every student's score. Question What effect would this change have on the distribution of the test scores?
An instructor was concerned that the highest score on an exam was 95, rather than 100, so they decided to add 5 points to every student's score. Question What effect would this change have on the distribution of the test scores?
The table below shows the results of 2 class sections that took a statistics exam and received either an A or B. What is true about the students who received an A and are in Section 1?
Both are dependent because P(from section 1|A grade) ≠ P(from section 1).
While trying to estimate the proportion of people who like to eat toast with butter, you keep a running total of how many have said yes and how many have said no. You notice that the proportion tends to change a lot. How do you ensure that you get a good estimate of the proportion who like to eat toast with butter?
Continue collecting observations to increase the sample size.
If the events X and Y are not disjoint, what is true?
Correct answer: P(X or Y) = P(X) + P(Y) - P(X and Y) Explanation Two events are disjoint when they do not have any outcomes in common:P(X and Y)=0 If the events are not disjoint, then:P(X and Y)≠0 Hence, if the events X and Y are not disjoint, then:P(X or Y)= P(X)+ P(Y) - P(X and Y)
Stratified sampling
Divides a population into subgroups (or strata). Random sampling is then used to identify a proportionate number of individuals from each subgroup (or stratum).
Considering a test of the null and alternative hypotheses, what statement is correct?
Exactly one hypothesis must be true.
What can reduce sampling errors in a survey?
Increasing the size of the population
Why might a two-tailed test be used to test a hypothesis instead of a one-tailed test?
It creates no expectation about the direction of effect.
A machine produces widgets. Each widget is produced independently and each has a mean diameter of 4.6 inches, with a standard deviation of 0.1 inches. A sample of 100 widgets is analyzed. What is the mean and standard deviation of the sample?
Mean 4.6 inches, standard deviation 0.01 Explanation From the Central Limit Theorem, the mean of the sample is the same as the mean for the individual events, and the standard deviation is the standard deviation of the individual events divided by the square root of the sample size.
Researchers often wish to reduce their sample size to save money in conducting studies. Question What factor might make a study's sample size smaller?
Reduce the nonresponse rate and stratify the population.
What method might be used to avoid bias in a clinical study?
Select patients randomly using rigorous criteria to avoid confounding results
How will two confidence intervals (90% and 98%) constructed from the same data compare?
The 98% confidence interval will be wider than the 90% confidence interval.
Is the Wilson Estimator biased or unbiased? And how does it behave under asymptotic (increasing sample size) conditions?
The Wilson Estimator is biased but asymptotically unbiased.
A Bernoulli random variable consists of a single experiment that can give a yes (1) or a no (0). The probability of a yes is `p`, and the probability of a no is `(1-p)`. You conduct an experiment in which the probability of a 1 is 70%. The probability of a 0 is 30%. What is the expected value of this experiment?
The expected value is the sum of the result (1) times the probability (0.7) plus (0) times (0.3). (1×0.7)+(0×0.3)=0.7.
The scatter plot shows the relation between two variables. Which of the following is true? Refer to the image.
The relation is strong and the slope is zero. Explanation The relation is strong because the dots are tightly clustered around a line. Across the entire scatterplot, the slope is zero. It is because the slope in the first half of the scatter plot is negative and in the second half is positive. In terms of magnitude, these slopes are equal. Thus, the slope of the entire scatter plot is zero.
What is true about the standard deviation of the mean?
The standard deviation of the mean decreases as n increases. This is because the standard deviation is calculated as the square root of the variance (that is, the average squared deviation from the mean).
When is the point estimator A a more efficient estimator for a given parameter than the point estimator B?
The variance of A is always less than the variance of B.
What is the most accurate description of the null hypothesis?
There are no significant observable differences between populations.
A group of doctors is studying the sleeping habits of nursing majors. They are wondering if the students sleep less soundly during finals week. The researchers collected data from 26 students about how many times they woke up during the middle of the night 2 weeks before finals and during the week of finals. The 95% confidence interval for the difference of the means between finals week and the control week is (-0.6, 1.2). Question What can the researchers conclude from their results?
There is likely no difference between the two time periods. In this case, the interval contains 0, which would mean there is no difference in the number of awakenings between the 2 weeks being compared. The results of this study are consistent, with there being no difference between the two time periods.
A group of doctors is studying the sleeping habits of nursing majors. They are wondering if the students sleep less soundly during finals week. The researchers collected data from 26 students about how many times they woke up during the middle of the night 2 weeks before finals and during the week of finals. The 95% confidence interval for the difference of the means is (-0.6, 1.2). Question What can the researchers conclude from their results?
They are 95% confident the true mean difference is between -0.6 and 1.2 awakenings.
The standard error is found by dividing the sample standard deviation by √n,
True
The linear regression equation has the following general form (without error): ˆy=b1x+b0 Which variable is a slope and what does it describe?
b1 represents the rate of change of ˆy for one unit increase in x.
Convenience sampling
is considered a non-random method because it is based on selecting individuals who are easily accessible.
Systematic sampling
uses random sampling to select by a fixed starting point in a larger population. Then a constant interval between samples is used to select the samples to include in the survey.
Cluster sampling
uses random sampling to select clusters and every individual in the chosen cluster is included in the sample.
