Probability
Sampling Distribution
Confidence Intervals
Significance Tests
Midterm 1
100
You roll two regular, four-sided dice. Write the probability model for the sum of the numbers of the two dice.
2: 1/16
3: 2/16
4: 3/16
5: 4/16
6: 3/16
7: 2/16
8: 1/16
100
Write the notation/symbols for the following terms: population mean, population standard deviation, sample mean, sample standard deviation, sampling distribution mean, sampling distribution standard deviation
100
A company want to estimate the mean number of hours employees work remotely each week. The sample mean is 18.4 hours, with a population standard deviation of 4.2, and n = 49. Construct a 95% confidence interval for the population mean.
(17.22, 19.58)
100
A researcher wants to test if the average amount of sleep college students get is different from 7 hours. Write the null and alternative hypothesis.
H0: μ = 7, Ha: μ ≠ 7
100
Use the 1.5xIQR rule to identify any suspected outliers in the dataset:
3, 4, 4, 7, 7, 7, 8, 10, 12, 50
[Q1-1.5IQR, Q3+1.5IQR] → [-5, 19]
200
A card is randomly selected from a standard deck of 52 cards. What is the probability that the card is a heart or a king?
13 hearts/52 + 4 kings/52 - 1/52 = 16/52
200
A student takes two random samples from the same population:
Sample A has 8 people, Sample B has 200 people.
Which sample mean is more likely to be closer to the true population mean, and why?
Sample B is more likely to be closer to the population mean → Law of Large Numbers, larger samples tend to produce sample means closer to the true population mean.
200
Suppose you want a smaller margin of error in a confidence interval. Which of the following would help?
1. Increase the sample size
2. Increase the confidence level
3. Decrease the population standard deviation.
Increase the sample size, decrease the population standard deviation
200
A coffee shop claims customers spend 15 minutes on average in line. You take a sample: x̄ = 17.2, μ0 = 15, σ = 6, n = 36. Find the test statistic.
z = 2.2
200
A student scores 85 on an exam where the mean is 75 and the standard deviation is 5. What is their z-score, and what does it represent?
z = 2
The student scored 2 standard deviations above the mean, meaning that the student scored fairly high relative to their class.
300
The time students spend studying for an exam is Normally distributed with a mean of 12 hours, and a standard deviation of 3 hours. Find the P(x < 9). Interpret the value in context.
15.87% of students study less than 9 hours.
300
The number of minutes people spend exercising each week is strongly right-skewed with a mean of 210, and a standard deviation of 60. A random sample of 100 people is selected. Find the probability that one randomly selected person exercises more than 250 minutes per week.
You cannot use central limit theorem to solve this because it only applies to the sampling distribution of the sample mean, not individual observations.
300
A researcher creates a 90% confidence interval for a population mean and gets: (42.1, 47.8). Which statement is correct?
a. 90% of the data values are between 42.1 and 47.8
b. There is a 90% chance the sample mean is between 42.1 and 47.8
c. We are 90% confident the population mean is between 42.1 and 47.8
d. 90% of all sample means must fall between 42.1 and 47.8
c. We are 90% confident the population mean is between 42.1 and 47.8
300
A p-value of 0.003 is found in a significance test. What does this p-value mean in context?
It means there is a 0.3% chance of getting a sample result this extreme (or more extreme) assuming the null hypothesis is true.
300
Given that x̄ = 12, sx = 3, ȳ = 32, sy = 9, and r = 0.45, find the Least-Squares Regression Line.
ŷ = 1.35x + 15.8
400
A student says:
“If I flip a fair coin 6 times in a row and get heads every time, the next flip is more likely to be tails.”
Is the student correct? Explain why or why not.
The student is NOT correct. Each coin flip is independent, even after getting 6 heads in a row, the probability of tails on the next flip is still 0.5.
400
The average amount of sleep college students get per night has a population mean of 6.8 hours, and a population standard deviation of 1.5 hours. A random sample of 49 students is taken. What is the probability that the sample mean sleep time is greater than 7.2 hours?
P(x̄ > 7.2) = 0.0307
400
Two students use the same data set to create confidence intervals for the population mean. Student A makes a 90% confidence interval. Student B makes a 99% confidence interval. Which interval will be wider, and why?
The 99% confidence interval will be wider because a higher confidence makes a larger margin of error, and a larger margin of error means a wider interval.
400
A gym claims the average person lifts 100 pounds on a certain machine. You collect data: x̄ = 94, μ0 = 100, σ = 12, n = 16, α = 0.05, Ha: μ < 100. Find the test statistic, p-value, and state your decision.
Test statistic = -2
p-value = 0.0228
0.0228 < 0.05 → reject H0
400
A regression line predicts a value of 52, but the actual observed value is 48. What is the residual, and what does its sign mean?
residual = observed y - predicted y = 48 - 52
residual = -4
The observed value was less than predicted, and our prediction was an overestimate.