Linear Regression
If two variables exhibit a correlation coefficient of -0.87, which of the following is a correct interpretation of the variables' relationship?
A. There is a weak negative linear relationship between the two variables
B. There is a moderate positive linear relationship between the two variables
C. There is a strong negative linear relationship between the two variables
D. The relationship between the variables is non-linear
What is C? Keep in mind that the range of r values is [-1, 1], so a correlation coefficient of -0.87 would indicate a strong, negative relationship.
Given that the random variable X represents the number of goals scored by a soccer team in a match, where X has has the following probability distribution:
P(X = 0) = 0.2
P(X = 1) = 0.5
P(X = 2) = 0.3
The expected value, E(X), is _____
What is 1.1? E(X) = 0 * 0.2 + 1 * 0.5 + 2 * 0.3 = 1.1
The time it takes for a bus to arrive at a particular stop follows a continuous uniform distribution between 5 and 15 minutes. P(7 < X < 12) = ___ and its interpretation is _____.
What is 0.5? 1 / (15 - 5) * (12 - 7) = 0.5
What is "There is a 50% chance that the bus arrives between 7 and 12 minutes?"
A researcher is studying the average number of hours college students spend monthly exercising. The population of all college students has a mean of 15 hours and a standard deviation of 4 hours. The researcher sampled 36 students, and 10.8% of them indicated their favorite day to exercise was on Sunday.
A. The standard deviation of X_bar is _____.
B. The standard deviation of p_hat is _____.
A. What is 2/3? = 4 / sqrt(36)
B. What is 0.05? = sqrt[(0.108 * (1 - 0.108)) / 36]
What is downward? Remember, square rooting the coefficient of determination only works to find the correlation coefficient when the regression line is upwards-sloping. If it downwards-sloping, then we have to add a negative sign to the square root of the coefficient of determination.
A factory produces light bulbs, and the probability that a bulb is defective is p = 0.05. Let X be the number of bulbs inspected before finding the first defective one. The probability that the first defective bulb is found on the 4th inspection is ____.
What is 0.0429? P(X = 4) = (1 - 0.05)3 * 0.05 = 0.0429
A factory produces machine parts, and the time to complete a part follows a normal distribution with a mean of 120 minutes and a standard deviation of 10 minutes. P(100 < X < 110) - P(X > 150) = ____
Answer with an Excel formula only if necessary.
What is 13.35?
P(100 < X < 110) = 47.5% - 34% = 13.5%
P(X > 150) = 0.15%
13.5% - 0.15% = 13.35%
A researcher is studying the average number of hours college students spend monthly exercising. The population of all college students has a mean of 15 hours and a standard deviation of 4 hours. The researcher sampled 36 students, and 10.8% of them indicated their favorite day to exercise was on Sunday.
A. The probability a randomly selected student exercises more than 17 hours is ____.
B. The probability that the true proportion of students who prefer exercising on Sundays is less than 8% is _____.
Answer with Excel formulas if applicable.
A. What is 1 - norm.dist(17, 15, .666, TRUE)?
B. What is norm.dist(0.08, 0.108, 0.05, TRUE)?
A student mistakenly flips the X and Y axes when creating a scatterplot. This would _____ the data's correlation coefficient.
What is not change? The correlation coefficient measures the strength and direction of the relationship between two variables, not the order in which they are plotted!
In a survey, 5 people are randomly chosen, and for each person, there is a 70% chance they support a certain candidate. Let X be the number of people who support the candidate out of the 5. The probability that exactly 3 people support the candidate is _____.
The heights of a population of adult males follows a normal distribution with a mean of 70 inches and a standard deviation of 3 inches. On a standard normal curve, a 75in-tall male would be _____ standard deviations _____ the mean.
What is 1.67 standard deviation above the mean?
z = (75 - 70) / 3 = 5/3
A. If you increase the sample size when creating a confidence interval, the margin of error becomes _____ and the confidence interval becomes ______.
B. If you increase the confidence level of a confidence interval, the margin of error becomes ______ and the confidence interval becomes ______.
A. What is "the margin of error becomes smaller and the confidence interval becomes narrower"? higher sample size --> lower MoE --> narrower CI
B. What is "the margin of error becomes larger and the confidence interval becomes wider"? higher confidence level --> higher critical value --> higher MoE --> wider CI
After performing a simple linear regression, you notice a quadratic (U-shaped) pattern in the residuals. What does this suggest about the model?
A. The model is a good fit since the residuals are scattered with inconstant variance
B. The model may be a poor fit since the residuals are scattered with constant variance
C. The model may be a poor fit since the residuals are scattered with inconstant variance
D. The model is a good fit since the residuals are scattered with constant variance
What is C? A pattern in the residuals is known as "inconstant variance", but we want the residuals to be scattered randomly (i.e., with "constant variance")
A company sells a product, and the probability that customer buys it is 0.3. Let X1 be the number of customers a company has to approach until the first one buys the product. Let X2 be the number of customers who buy the product in a group of 10 customers.
P(X2 = 3) - P(X1 = 4) = _____
What is 0.1639?
P(X2 = 3) = (10 choose 3) * (0.3)3 * (0.7)7 = 0.2668
P(X1 = 4) = (1 - 0.3)3 * 0.3 = 0.1029
0.2668 - 0.1029 = 0.1639
The population of annual incomes of a certain city has a mean of $45,000 and a standard deviation of $12,000.
A. The probability that a randomly selected resident of the city makes more than $58,000 is _____.
B. The dollar amount that corresponds to the 88% percentile of annual income is ______.
A. What is 1 - norm.dist(58000, 45000, 12000, TRUE)?
B. What is norm.inv(0.88, 45000, 12000)?
Joey wants to conduct a study over the population proportion of Iowa students who know how to use the Transit app. He wants to create a confidence interval at 90% confidence with a range of 6%. If the sample proportion of Iowa students who don't know how to use the Transit app is 0.82, Joey should sample _____ students.
What is 444?
p_hat = 1 - 0.82 = 0.18 | CV for 90% CI = 1.645 | MoE = 6% / 2 = 3%
N = [0.18 * (1 - 0.18) * 1.6452] / 0.032
Let X ~ N(15, sigma) and P(X < 18.75) = 0.9985.
sigma = ____
x = 18.75 | z = 3 (shown by the 0.9985 percentile)
3 = (18.75 - 15) / sigma
sigma = 1.25
Of a sample of 345 Statistics for Business students, 36% said Probability was the hardest unit to understand. Construct a 90% confidence interval for the population proportion of Statistics for Business students who believe Probability is the hardest unit to understand.
A. The confidence interval is (____, ____)
B. If you wanted to create the interval at 74% confidence, the critical value would be _____. Answer with an Excel formula if applicable.
A. What is (0.317, 0.403)?
= 0.36 +/- 1.645 * sqrt[(0.36 * (1 - 0.36)) / 345]
B. What is norm.inv(0.13, 0, 1)?
Alpha = 1 - 0.74 = 0.26 | Alpha / 2 = 0.26 / 2 = 0.13