Understanding Sampling, Estimation, and Hypothesis Testing in Statistics

Selecting Samples

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

A random sample of size n from an infinite population is a sample selected such that the following conditions are satisfied.

Point Estimation

By making the preceding computations, we perform the statistical procedure called point estimation. We refer to the sample mean x as the point estimator of the population mean m, the sample standard deviation s as the point estimator of the population standard deviation s, and the sample proportion p as the point estimator of the population proportion p. The numerical value obtained for x, s, or p is called the point estimate.

Sampling Distributions

The sampling distribution of x is the probability distribution of all possible values of the sample mean x. Because the sample mean x is a quantity whose values are not known with certainty, the sample mean x is a random variable. As a result, just like other random variables, x has a mean or expected value, a standard deviation, and a probability distribution. Because the various possible values of x are the result of different simple random samples, the probability distribution of x is called the sampling distribution of x. Knowledge of this sampling distribution and its properties will enable us to make probability statements about how close the sample mean x is to the population mean m.

Interval Estimation

Interval estimation is frequently used to generate an estimate of the value of a population parameter. An interval estimate is often computed by adding and subtracting a value, called the margin of error, to the point estimate: Point estimate +/- Margin of error

Hypothesis Testing

In hypothesis testing we begin by making a tentative conjecture about a population parameter. This tentative conjecture is called the null hypothesis and is denoted by H0. We then define another hypothesis, called the alternative hypothesis, which is the opposite of what is stated in the null hypothesis. The alternative hypothesis is denoted by Ha. The hypothesis testing procedure uses data from a sample to test the validity of the two competing statements about a population that are indicated by H0 and Ha.

If H0 is true, this conclusion is correct. However, if Ha is true, we made a Type II error; that is, we accepted H0 when it is false. The second row of Table 6.6 shows what can happen if the conclusion is to reject H0. If H0 is true, we made a Type I error; that is, we rejected H0 when it is true. However, if Ha is true, rejecting H0 is correct.

30. a. H0: μ ≥ 220 Ha: μ < 220
b. Claiming μ < 220 when the new method does not lower costs. This could lead to implementing the method when it does not lower costs.

Gasoline prices were relatively steady for about the first 16 to 18 months and then increased rapidly through about month 25 before falling before rising in the last few months. Overall the price of gasoline appears to be increasing over the 36 months, but it is not a constant increase.