Question bank
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Variance of a sample mean
A population has expectation \mu = 50 and variance \sigma^2 = 36. A random sample of size n = 9 is drawn, and \bar{X} is its sample mean.
Standard error of the mean
Battery lifetimes have standard deviation \sigma = 12 hours. A quality engineer averages the lifetimes of n = 36 randomly chosen batteries.
Sum of a sample, right tail
X_1, \dots, X_{25} are independent, each normal with expectation \mu = 4 and variance \sigma^2 = 9. Let S = \sum_{i=1}^{25} X_i.
Sample mean, left tail
The weight of a bag of flour is normal with \mu = 200 g and \sigma = 20 g. An inspector weighs n = 16 bags and computes their mean \bar{X}.
Variance of a scaled sum
X_1, \dots, X_{10} are independent and identically distributed with Var(X_i) = 5. Let Y = 2\sum_{i=1}^{10} X_i.
Standardizing, far right tail
A reaction time in a test is normal with expectation \mu = 70 ms and variance \sigma^2 = 25.
Probability of an interval
X \sim N(10, 4), where 4 is the variance.
Total of many deliveries (CLT)
A courier's delivery times are independent, with expectation \mu = 30 minutes and standard deviation $\sigma = 10…
Sample size for a target standard error
A measurement has standard deviation \sigma = 6. We want the standard error of the sample mean to be at most d = 0.5.
Scaled sum against a threshold
X_1, X_2, X_3, X_4 are independent, each N(25, 16) (16 is the variance). Let T = 5\sum_{i=1}^{4} X_i.
Chebyshev bound
A random variable X has expectation \mu = 50 and standard deviation \sigma = 5; nothing else is known about its distribution.