Final Practice Exam

Suppose resting heart rates of healthy adults follow a normal model with mean μ = 72 beats per minute and standard deviation σ = 8 beats per minute. Find the z-score of a resting rate of 56 beats per minute. Type a whole number, including its sign.

Type a short answer.

A friend insists that she picked answers at random on a 10-question trivia sheet with four choices per question, yet she got 6 right. If she is guessing, her number correct X is binomial with n = 10 and p = 0.25, and a table shows P(X = 6) = 0.0162, P(X = 7) = 0.0031, P(X = 8) = 0.0004, P(X = 9) = 0.0000 and P(X = 10) = 0.0000, each to four decimals. Find the p-value P(X ≥ 6), rounded to four decimals.

Type a short answer.

Suppose each of 20 randomly chosen shoppers independently uses a coupon with probability p = 0.3, and X counts the coupon users, so X is binomial with n = 20. What are the binomial mean μ and standard deviation σ of X?

Select one.

• μ = 6 and σ ≈ 2.0494
• μ = 6 and σ ≈ 4.2
• μ = 14 and σ ≈ 2.0494
• μ = 20 and σ ≈ 2.0494

A student claims to have guessed on all 10 four-option questions of a quiz but scored 8. Under H₀: p = 0.25, the p-value is P(X ≥ 8) = 0.0004. Which sentence interprets this p-value correctly?

Select one.

• If the student were only guessing, a score of 8 or higher would occur on about 0.04% of such quizzes.
• The chance that the student was only guessing on this particular quiz is about 0.04%.
• The chance that the score of 8 came from luck alone is about 0.04%.
• The student knows far more than a guesser, because 0.0004 is a tiny number.

A campus wellness survey records the following for each student who responds. Which of these survey variables are categorical? Select all that apply.

Select every option that applies.

• blood type
• resting heart rate (beats per minute)
• home state
• hours of sleep last night
• class standing (first-year, sophomore, junior, senior)

A two-sample t procedure comparing reaction times in two groups prints 37.84 degrees of freedom. True or false: the output must be wrong, since degrees of freedom can only be whole numbers.

True or false.

• True
• False

The sum of two fair dice has population mean μ = 7 and population standard deviation σ = 2.4152. A player rolls the pair 40 times and averages the 40 sums. Find the standard error of that average, σ ⁄ √n, rounded to four decimal places.

Type a short answer.

A 95% one-proportion z interval for the share of shoppers who use a coupon runs from 0.3082 to 0.4278. Which statement interprets it correctly?

Select one.

• We are 95% confident that the proportion of all shoppers who use a coupon is between 0.3082 and 0.4278.
• The probability is 95% that the true proportion of coupon users is between 0.3082 and 0.4278.
• About 95% of the shoppers use between 0.3082 and 0.4278 of a coupon each.
• About 95% of future sample proportions will lie between 0.3082 and 0.4278.

A pollster has no prior estimate of the share and wants a 95% margin of error no larger than 4 percentage points. Using the conservative p̂ = 0.5 and z* = 1.96, what is the smallest number of people the poll should include? Type a whole number.

Type a short answer.

For coin flips and die rolls, which pairs of events are independent, so that knowing one event happened leaves the chance of the other unchanged? Select all that apply.

Select every option that applies.

• Two coin flips: "the first flip is heads" and "both flips are heads"
• One coin flip and one die roll: "the coin lands heads" and "the die shows 3"
• One die roll: "the roll is even" and "the roll is odd"
• Two coin flips: "the first flip is heads" and "exactly one flip is heads"

Researchers weighed 123 Gentoo penguins and found a sample mean of x̄ = 5,076.02 g with a sample standard deviation of s = 504.12 g. Which value serves as the point estimate of μ, the mean mass of all Gentoo penguins like these?

Select one.

• s = 504.12 g
• The standard error, 45.45 g
• x̄ = 5,076.02 g
• μ itself, which the sample shows exactly

For the daily step counts of 40 volunteers, a spreadsheet's QUARTILE.INC returns Q1 = 5,200 steps and Q3 = 8,600 steps. Using those two values, what is the upper fence of the 1.5 × IQR rule, in steps? Enter the number only.

Type a short answer.

A wellness coach asks whether adults who sleep more hours tend to report lower stress scores. On a scatterplot of 85 adults, which variable is the explanatory variable, and on which axis does it belong?

Select one.

• Hours of sleep, on the horizontal axis, because the question starts from it
• Stress score, on the horizontal axis, because it is the variable being explained
• Hours of sleep, on the vertical axis, because it is recorded first
• Stress score, on the vertical axis, because higher stress causes less sleep

Which of these statements about the normal curve, used as a model for data, are correct? Select all that apply.

Select every option that applies.

• Exactly 2 numbers pin it down: the mean μ sets its center and the standard deviation σ its spread
• A real-world quantity such as adult height may be called normal as a stated fact, with no check
• Its total area, from the far left to the far right, equals 1
• The height of the curve at one exact value is the probability of that value
• It is symmetric, with its single peak at the mean μ

A city survey's 95% interval for the share of households that keep a garden runs from 0.2860 to 0.3540. True or false: because 0.3 falls inside this interval, the survey shows that exactly 30% of the city's households keep a garden.

True or false.

• True
• False

A sample of five commute distances, in miles, is 4, 7, 9, 10, and 15. Their mean is 9 miles, and their squared deviations from the mean add to 66. What is the sample variance s², in square miles? Enter the number only.

Type a short answer.

A random sample of 25 bags of coffee beans has a mean weight of x̄ = 498.6 g and a standard deviation of s = 12.5 g, so the standard error is 2.5 g. The 95% critical value for df = 24 is t* = 2.064. How large is the margin of error in grams? Give the number to two decimal places without units.

Type a short answer.

A spinner is supposed to land on blue one quarter of the time. In 400 spins it lands on blue 112 times, and you test H₀: p = 0.25 against Hₐ: p ≠ 0.25 with a one-proportion z test. Which standard error does this test use?

Select one.

• √(p̂(1 − p̂) ⁄ n), using the sample share p̂
• √(p₀(1 − p₀) ⁄ n), using the null value p₀ = 0.25
• s ⁄ √n, using the sample standard deviation s
• √(p̂_pool(1 − p̂_pool)(1 ⁄ n₁ + 1 ⁄ n₂))

Suppose a delivery robot completes each run successfully with probability p = 0.7, independently, and X counts its successes in n = 5 runs. Using the binomial probability formula, what is P(X = 3), rounded to four decimals?

Select one.

• 0.0309
• 0.7000
• 0.1323
• 0.3087

For 48 houses, the regression of monthly heating cost on floor area has r² = 0.64. Which statement interprets r² correctly?

Select one.

• About 64% of the 48 houses sit exactly on the least-squares line.
• Floor area causes about 64% of every house's heating cost.
• Each extra square foot predicts 0.64 dollars more in heating cost.
• Floor area accounts for about 64% of the variation in monthly heating cost.

Suppose single battery lifetimes follow a normal model with μ = 520 hours and σ = 40 hours. The average lifetime of a random pack of 16 batteries has a standard error of 10 hours. Roughly what share of 16-battery averages exceed 530 hours?

Select one.

• About 16%
• About 32%
• About 2.5%
• About 34%

A bakery weighs a random sample of 10 loaves: x̄ = 452.30 g and s = 11.42 g. With SE = 3.61 g and t* = 2.262 for 95% confidence at df = 9, which is the 95% t interval for the mean loaf weight μ?

Select one.

• 426.47 g to 478.13 g
• 448.69 g to 455.91 g
• 445.22 g to 459.38 g
• 444.13 g to 460.47 g

A sports analyst finds a correlation of r = 0.62 between players' heights and their rebounds per game. Select every true statement about this correlation coefficient.

Select every option that applies.

• Measuring height in centimeters instead of inches would leave r at 0.62
• A single extreme player could change r noticeably
• An r of 0.62 means that 62% of the players follow the pattern
• Any correlation, including this one, must lie between −1 and 1
• The same kind of r could measure how team name relates to position

A spinner has four equal sections labeled 2, 4, 6, and 9, and X is the label it lands on, each with probability 1 ⁄ 4. What is the expected value E(X)?

Select one.

• 21
• 4
• 5.25
• 5

A radio host invites listeners to phone in and say whether the city should build a new stadium, and thousands of callers respond. Which statement about the callers' opinions is correct?

Select one.

• They are accurate, because so many callers cancel out any bias
• They form a simple random sample, because anyone at all was free to call in
• They are a parameter, because they include every listener's view
• They can still be biased, because people who feel strongly choose to call

Text messages went to a random sample of 1,500 registered voters, and the survey reports a 95% margin of error of about 2.5 percentage points. Which sources of error lie outside what this margin of error covers? Select all that apply.

Select every option that applies.

• Voters with no mobile phone, who could never be reached by text
• The luck of which 1,500 voters happened to be chosen at random
• A question phrased in a way that nudges voters toward one answer
• The sample-to-sample variation in p̂ that would shrink with a bigger sample
• Chosen voters who never reply and who hold different views from those who do

An inspector finds that every one of 80 weighed cereal boxes sits within three standard deviations of the boxes' mean weight. True or false: that single finding is enough to establish that the box weights follow a normal distribution.

True or false.

• True
• False

A coin is flipped 20 times and lands heads 13 times. A test of H₀: p = 0.5 against Hₐ: p > 0.5 gives a p-value of 0.1316, so it fails to reject H₀ at α = 0.05. True or false: this result establishes that the coin is fair.

True or false.

• True
• False

A field team recorded the body masses of 68 Chinstrap penguins: x̄ = 3,733.09 g and s = 384.34 g. What degrees of freedom go with a one-sample t interval for their population mean?

Select one.

• 67
• 68
• 69
• 66

On an easy quiz taken by 180 students, the mean score is 81.4 points and the median is 85 points. The first quartile is 76 and the third quartile is 90, so the median sits 9 points above Q1 but only 5 points below Q3. What do these summaries suggest about using a normal model for the scores?

Select one.

• A normal model fits well, because the 1.5 × IQR rule flags none of the quiz scores
• A normal model fits well, because the mean and the median both sit near 83 points
• A normal model is a poor fit for detailed tail questions, because the scores lean right
• A normal model is a poor fit for detailed tail questions, because the scores lean left

A delivery company's daily package count, C, has mean 180 and standard deviation 12. A dispatcher logs C − 150, how far each day's count lands above 150. What are the mean and the standard deviation of C − 150?

Select one.

• Mean 180 and SD 12
• Mean 30 and SD 0
• Mean 30 and SD 144
• Mean 30 and SD 12

Software compares mean nightly sleep for 85 night-shift and 92 day-shift workers and prints: difference in means 0.74 h, standard error of the difference 0.21 h, Welch t = 3.524, about 171.3 degrees of freedom, and a p-value below 0.001. Which printed value tells how many standard errors separate the two sample means?

Select one.

• The difference in means, 0.74 h
• The standard error, 0.21 h
• The degrees of freedom, 171.3
• The Welch t statistic, 3.524

A television station invited viewers to text in their vote, and 25,000 texts arrived, so the conservative 95% margin of error is under 1 percentage point. Which statement is correct?

Select one.

• The margin already corrects for the many viewers who watched but chose not to text in.
• The result is within about 1 percentage point of the true share of all viewers.
• The margin reflects only random sampling variation, so self-selection bias could push the result far off.
• Collecting 50,000 texts instead would get rid of the bias in the vote.

A coffee shop records each order's milk choice: whole, oat, almond, or no milk. Which measure of center can describe this variable?

Select one.

• The mean, after coding the choices as 1, 2, 3, and 4
• The median, because the choices can be listed alphabetically
• The mean, because every order has a milk choice
• The mode, because milk choice is a nominal category

Another student who says he guessed gets 2 of 10 right on a quiz with four options per question. Under the guessing model (n = 10, p = 0.25), P(X ≤ 1) = 0.2440 and P(X = 2) = 0.2816. What is the p-value P(X ≥ 2), rounded to two decimals?

Select one.

• 0.24
• 0.28
• 0.76
• 0.53

Under this course's rules, for which of these statistics is a normal model for the sampling distribution reasonable? Select all that apply.

Select every option that applies.

• The sample proportion p̂ in a random survey of 400 households when p = 0.3, so np = 120 and n(1 − p) = 280
• The average lifetime of 9 batteries when single lifetimes are assumed to follow a normal model with μ = 520 hours and σ = 40 hours
• The proportion of correct answers when guessing on 10 four-option questions, where np = 10 × 0.25 = 2.5
• The average of 40 rolls of one fair die, whose six faces are equally likely
• The 50 separate sums in a single sample of 50 rolls of two fair dice

At one clinic site, 38 of 120 patients missed a follow-up visit; at a second site, 61 of 140 patients missed one. Which procedure tests whether the miss rate differs between the two sites?

Select one.

• A one-proportion z test
• A two-sample t test
• A two-proportion z test
• A one-sample t test

Suppose the weights of apples picked in one orchard follow a normal model with mean μ = 150 g and standard deviation σ = 12 g. By the empirical rule, between which two weights do about 95% of apples fall?

Select one.

• From 138 g to 162 g
• From 126 g to 174 g
• From 114 g to 186 g
• From 144 g to 156 g

Suppose delivery times for a meal service follow a normal model with mean μ = 45 minutes and standard deviation σ = 8 minutes. The z-value with 95% of the standard normal area to its left is 1.645. Only 5% of deliveries take longer than which time?

Select one.

• 31.84 minutes
• 53.00 minutes
• 60.68 minutes
• 58.16 minutes

A student who says she guessed scores 4 of 10 on a quiz with four options per question. Under H₀: p = 0.25 the p-value is P(X ≥ 4) = 0.2241, and the significance level fixed beforehand is α = 0.10. Which decision and conclusion follow?

Select one.

• Reject H₀; the score shows she was not guessing.
• Fail to reject H₀; a score this high is consistent with guessing.
• Accept H₀; the score proves she was guessing.
• Reject H₀, since 4 correct is above the guessing mean of 2.5.