Statistics (AP-aligned)

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  1. A survey asked 145 people whether they exercise weekly. Under 30: 22 yes, 43 no. Age 30 or over: 27 yes, 53 no. What proportion of the under-30s exercise weekly?
  2. A researcher measures the sleep of 376 randomly chosen teenagers and finds a mean of 7.1 hours. What is 7.1?
    1. A statistic: it describes the sample
    2. A parameter: it describes all teenagers
    3. The population
  3. Two fair dice are rolled. Find the probability that the sum is 2.
  4. A random variable X takes the values 0, 1 and 2, with P(X = 0) = 29/100 and P(X = 1) = 13/100. Find P(X = 2).
  5. Many random samples of size 161 are taken from the same population. Which quantity changes from sample to sample?
    1. The sample proportion p̂ (a statistic)
    2. The population proportion p (a parameter)
    3. The sample size
  6. A 95% confidence interval for a proportion is (39/100, 49/100). Find the margin of error.
  7. A company claims that 80% of customers are satisfied. A researcher suspects fewer are. Which is the null hypothesis?
    1. H₀: p = 0.80
    2. H₀: p < 0.80
    3. H₀: p̂ = 0.80
  8. A regression on a random sample of 23 houses gives slope b. What does b estimate?
    1. The slope β₁ of the population regression line
    2. Nothing: b is the true slope
    3. The correlation r
  9. A distribution of incomes has mean 54 thousand dollars and median 43 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric
  10. Of 40 drivers who texted, 30 had a near-miss; of 40 who did not text, 10 had one. Is there an association between texting and near-misses in these data?
    1. Yes: 75% of texters vs 25% of non-texters had a near-miss
    2. No: both groups have 40 drivers
    3. Yes, and it proves texting causes near-misses
  11. A school has 600 juniors and 300 seniors. A stratified sample of 90 students is taken in proportion to grade. How many juniors are in the sample?
  12. In a class of 35 students, 12 are girls, 12 play soccer and 2 girls play soccer. A student is chosen at random. Find P(girl or soccer player).
  13. X takes the values 0, 1, 2 with probabilities 51/100, 15/100 and 34/100. Find E(X).
  14. The population proportion is p = 8/10. Find the standard deviation of p̂ for random samples of size 83.
  15. In a random sample of 400, p̂ = 5/10. Using z* = 1.96, find the margin of error of a 95% interval for p.
  16. Test H₀: p = 0.5 with a random sample of 100 in which p̂ = 52/100. Find the z statistic.
  17. Computer output gives slope b = 8/10 with standard error SE_b = 3/100. Find the t statistic for H₀: β₁ = 0.
  18. Find the mean of the data 5, 11, 24, 33, 41.
  19. A scatterplot of car age against resale value has r=−0.94r = -0.94. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value
  20. A news website asks readers to vote online; 4824 vote and 80% favor a new law. What is the main problem?
    1. Voluntary response bias: people with strong opinions choose to respond
    2. The sample is too small
    3. None: 4824 responses are representative
  21. A screening test was given to 316 people: 47 had the disease and tested positive, 8 had the disease and tested negative, 34 did not have it and tested positive, 227 did not have it and tested negative. Find P(disease | positive).
  22. X and Y are independent with SD(X) = 27 and SD(Y) = 36. Find SD(X + Y).
  23. A population has standard deviation σ = 9. Find the SD of the sample mean for samples of size 49.
  24. From the same sample of 458, a 99% and a 90% interval are computed. Which is wider?
    1. The 99% interval
    2. The 90% interval
    3. They have the same width
  25. At significance level α = 0.05, a test gives P-value 0.04. What do you conclude?
    1. Reject H₀: there is convincing evidence for Hₐ
    2. Accept H₀
    3. Hₐ is proven true
  26. With 30 pairs, b = 2/10 and SE_b = 7/100. Using t* = 2.048 (28 df, 95%), find the upper end of the interval for β₁.
  27. Find the interquartile range (IQR) of the 8 values 3, 9, 14, 17, 24, 27, 33, 46.
  28. r=0.6r = 0.6, sx=15s_x = 15 and sy=16s_y = 16. Find the slope of the least-squares regression line of yy on xx.
  29. Researchers randomly assign 66 volunteers to drink coffee or water, then test their reaction times. What kind of study is it?
    1. An experiment: the researchers assign the treatment
    2. An observational study
    3. A sample survey
  30. Two independent machines fail with probabilities 0.8 and 0.7. Find the probability that both fail.
  31. A free throw is made with probability 1/6. In 5 independent shots, find P(exactly one is made).
  32. Household incomes are strongly right-skewed. What is the shape of the sampling distribution of x̄ for random samples of 52 households?
    1. Approximately normal, by the central limit theorem
    2. Right-skewed like the population
    3. Uniform
  33. Using p = 0.5 and z* = 1.96, find the value of n (before rounding up) needed for a margin of error of 0.04.
  34. Test H₀: μ = 39. A random sample of 9 has x̄ = 43 and s = 9. Find the t statistic.
  35. To test whether x is useful for predicting y (from 51 pairs), what is the null hypothesis?
    1. H₀: β₁ = 0, no linear relationship in the population
    2. H₀: b = 0
    3. H₀: r = 1
  36. A data set has Q1=24Q_1 = 24 and Q3=32Q_3 = 32. Above which value is a data point an outlier by the 1.5×IQR1.5 \times \text{IQR} rule?
  37. y^=30+3x\hat{y} = 30 + 3x. A data point is (7,46)(7, 46). Find its residual.
  38. Why are 196 subjects assigned to the treatments at random?
    1. To balance other variables between the groups, so differences can be blamed on the treatment
    2. So the results apply to the whole population
    3. To give every subject the same chance of being in the study
  39. 12% of a population has a condition. A test is positive for 86% of people with the condition and for 11% of people without it. Find P(positive).