The quantity sigma^2 / n, found by dividing the population's squared standard deviation by the sample size, is the variance of the sample mean.
Missing Word
variance
Variance is the square of a standard deviation. The variance of x-bar is sigma^2 / n, and its square root is the standard error sigma / sqrt(n).
Category: Standard Error
Question 2
A count of successes takes only whole numbers, but the normal distribution is continuous, so a 0.5 adjustment is made when one approximates the other.
Missing Word
continuous
A continuous distribution can take any value in an interval. Because a whole-number count has gaps between values, the continuity correction widens each count by 0.5 on each side before using the normal curve.
Category: Normal Approximation to Binomial
Question 3
Each trial counted by B(n, p) has exactly 2 possible outcomes, called success and failure.
Missing Word
2
A binomial experiment is made of independent trials, each with exactly two outcomes, success with probability p and failure with probability q = 1 - p.
Category: Normal Approximation to Binomial
Question 4
Income data usually have a long tail toward high values, so they are described as skewed to the right.
Missing Word
skewed
A skewed distribution is lopsided, with a longer tail on one side. Even when a population is skewed, the central limit theorem says sample means from large samples are approximately normal.
Category: Distribution Types
Question 5
A distribution in which every value between 1 and 5 is equally likely, written U(1, 5), is a uniform distribution.
Missing Word
uniform
A uniform distribution has a flat density, so equal-width intervals have equal probability. U(1, 5) has mean (1 + 5) / 2 = 3, and averages from it become approximately normal by the central limit theorem.
Category: Distribution Types
Question 6
The distribution often used for waiting times, whose standard deviation equals its mean, is the exponential distribution.
Missing Word
exponential
The exponential distribution is skewed right, and its mean and standard deviation are equal. Sample means from it still become approximately normal for large samples.
Category: Distribution Types
Question 7
The central limit theorem assumes the values in each sample are independent of one another, so one value tells you nothing about the next.
Missing Word
independent
Independent values do not influence each other. Random sampling provides this, and the standard error formula sigma / sqrt(n) relies on it.
Category: Sampling Distribution of Means
Question 8
The value below which 90% of all sample means fall is the 90th percentile of the sampling distribution of x-bar.
Missing Word
percentile
A percentile is the value with a given percentage of the distribution at or below it. For sample means it is found as mu plus a z-score times the standard error.
Category: Z-Scores and Percentiles
Question 9
When np and nq are both large enough, a binomial distribution of successes in n trials can be approximated by a bell-shaped curve with mean np.
Missing Word
binomial
The binomial distribution counts successes in a fixed number of independent trials with the same probability p. For large n with p not near 0 or 1, it is well approximated by a normal curve with mean np and standard deviation sqrt(npq).
Category: Normal Approximation to Binomial
Question 10
The population mean mu is a parameter, a fixed number that describes the entire population.
Missing Word
parameter
A parameter is a numerical summary of a whole population, usually unknown. Sample means are used to learn about it, and the central limit theorem describes how they vary around it.
Category: Sampling Distribution of Means
Question 11
The probability that x-bar falls between two values equals the area under its curve between those values.
Missing Word
area
For a continuous distribution, probability is measured as area under the density curve, and the total area under the curve is 1. Once the central limit theorem gives x-bar an approximately normal curve, a probability such as P(48 < x-bar < 52) is the area between 48 and 52 under that curve.
Category: Z-Scores and Percentiles
Question 12
When x-bar is approximately normal, about 99.7% of sample means fall within 3 standard errors of mu.
Missing Word
3
By the empirical rule, about 99.7% of a normal distribution lies within three standard deviations of the center. For sample means, the standard deviation is the standard error.
Category: Z-Scores and Percentiles
Question 13
Sampling with replacement means each selected individual is returned to the population before the next draw.
Missing Word
replacement
With replacement, every draw is made from the same full population, so the selections do not affect one another. This is one way to guarantee the separate draws the theorem assumes.
Category: Sampling Distribution of Means
Question 14
A count that can take only separate whole-number values, such as the number of heads in 10 coin tosses, is a discrete random variable.
Missing Word
discrete
A discrete random variable takes countable, separated values. A normal curve can still approximate such a count when n is large, using a 0.5 correction at each value.
Category: Normal Approximation to Binomial
Question 15
For large samples, the sampling distribution of x-bar is approximately normal, with its line of symmetry at mu.
Missing Word
symmetry
A line of symmetry splits a distribution into two mirror-image halves. The central limit theorem says sample means from large samples follow an approximately normal curve, and a normal curve is mirrored about its mean, mu.
Category: Sampling Distribution of Means