A review of key statistical concepts

An overview of the review Populations and parameters Samples and statistics Confidence intervals Hypothesis testing

Populations and Parameters … and Samples and Statistics

Populations and Parameters A population is any large collection of objects or individuals, such as Americans, students, or trees about which information is desired. A parameter is any summary number, like an average or percentage, that describes the entire population.

Parameters Examples: –population mean µ = average temperature –population proportion p = proportion approving of president’s job performance ….% of the time, we don’t (...or can’t) know the real value of a population parameter. Best we can do is estimate the parameter!

Samples and Statistics A sample is a representative group drawn from the population. A statistic is any summary number, like an average or percentage, that describes the sample.

Statistics Examples –sample mean (“x-bar”) –sample proportion (“p-hat”) Because samples are manageable in size, we can determine the value of statistics. We use the known statistic to learn about the unknown parameter.

Example: Smoking at PSU? Population of 42,000 PSU students What proportion smoke regularly? Sample of 987 PSU students 43% reported smoking regularly

Example: Grade inflation? Population of 5 million college students Is the average GPA 2.7? Sample of 100 college students How likely is it that 100 students would have an average GPA as large as 2.9 if the population average was 2.7?

Example: A linear relationship?

Two ways to learn about a population parameter Confidence intervals estimate parameters. –We can be 95% confident that the proportion of Penn State students who have a tattoo is between 5.1% and 15.3%. Hypothesis tests test the value of parameters. –There is enough statistical evidence to conclude that the mean normal body temperature of adults is lower than 98.6 degrees F.

Confidence intervals A review of concepts

The situation Want to estimate the actual population mean . But can only get “x-bar,” the sample mean. Use “x-bar” to find a range of values, L<  <U, that we can be really confident contains . The range of values is called a “confidence interval.”

Confidence intervals for proportions in newspapers “Sample estimate”: 69% of 1,027 U.S. adults think using a hand-held cell phone while driving a car should be illegal. The “margin of error” is 3%. The “confidence interval” is 69% ± 3%. We can be really confident that between 66% and 72% of all U.S. adults think using a hand-held cell phone while driving a car should be illegal. Source: ABC News Poll, May 16-20, 2001

General form of most confidence intervals Sample estimate ± margin of error Lower limit L = estimate - margin of error Upper limit U = estimate + margin of error Then, we’re confident that the value of the population parameter is somewhere between L and U.

(1-α)100% t-interval for population mean  Formula in notation: Formula in words: Sample mean ± (t-multiplier × standard error)

Determining the t-multiplier

Typical t-multipliers Conf. coefficientConf. level % % %0.995

t-interval for mean in Minitab One-Sample T: FVC Variable N Mean StDev SE Mean 95.0% CI FVC (3.4655,3.7095) We can be 95% confident that the mean forced vital capacity of all female college students is between 3.5 and 3.7 liters.

Length of confidence interval Want confidence interval to be as narrow as possible. Length = Upper Limit - Lower Limit

How length of CI is affected? As sample mean increases… As the standard deviation decreases… As we decrease the confidence level… As we increase sample size …

Hypothesis testing A review of concepts

General idea of hypothesis testing Make an initial assumption. Collect evidence (data). Based on the available evidence (data), decide whether to reject or not reject the initial assumption.

Example: Normal body temperature Population of many, many adults Is average adult body temperature 98.6 degrees? Or is it lower? Sample of 130 adults Average body temperature of 130 sampled adults is degrees.

Making the decision It is either likely or unlikely that we would collect the evidence we did given the initial assumption. If it is likely, then we “do not reject” our initial assumption. There is not enough evidence to do otherwise.

Making the decision (cont’d) If it is unlikely, then: –either our initial assumption is correct and we experienced a very unusual event –or our initial assumption is incorrect In statistics, if it is unlikely, we “reject” our initial assumption.

Again, idea of hypothesis testing: criminal trial analogy First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”) –H 0 : Defendant is not guilty (innocent). –H A : Defendant is guilty.

Criminal trial analogy (continued) Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. In statistics, the data are the evidence.

Criminal trial analogy (continued) Then, make initial assumption. –Our criminal justice system is based on “defendant is innocent until proven guilty.” –So, assume defendant is innocent. In statistics, we always assume the null hypothesis is true.

Criminal trial analogy (continued) Then, make a decision based on the available evidence. –If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis. (Behave as if defendant is guilty.) –If there is insufficient evidence, do not reject the null hypothesis. (Behave as if defendant is innocent.)

Very important point If we reject the null hypothesis, we do not prove the alternative hypothesis is true. If we do not reject the null hypothesis, we do not prove the null hypothesis is true. We merely state there is enough evidence to behave one way or the other. Always true in statistics! Whatever the decision, there is always a chance we made an error.

Errors in criminal trials

Errors in hypothesis testing

Definitions: Types of errors Type I error: The null hypothesis is rejected when it is true. Type II error: The null hypothesis is not rejected when it is false. There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the decision “It is either likely or unlikely that we would collect the evidence we did given the initial assumption.” Two ways to determine likely or unlikely: –Critical value approach (many textbooks) –P-value approach (science, journals, software)

Possible hypotheses about mean µ TypeNullAlternative Right-tailed Left-tailed Two-tailed

Critical value approach Using sample data and assuming null hypothesis is true, calculate the value of the test statistic. Set the significance level, α, the probability of making a Type I error to be small (0.05 or 0.01). Compare the value of the test statistic to the known distribution of the test statistic. If the test statistic is more extreme than expected, allowing for an α chance of error, reject the null hypothesis. Otherwise, don’t reject the null.

Right-tailed critical value Reject null hypothesis if test statistic is greater than

Left-tailed critical value Reject null hypothesis if test statistic is less than

Two-tailed critical value Reject null hypothesis if test statistic is less than or greater than

P-value approach Using sample data and assuming null hypothesis is true, calculate the value of the test statistic. Using known distribution of the test statistic, calculate the P-value = “If the null hypothesis is true, what is the probability that we’d observe a more extreme test statistic than we did?” Set the significance level, α, the probability of making a Type I error to be small (0.05 or 0.01). If the probability is small, i.e., smaller than α, reject the null hypothesis. Otherwise, don’t reject the null.

Right-tailed P-value If it’s unlikely to observe such a large test statistic, i.e., if the P- value (0.0127) is smaller than α, reject the null hypothesis.

Left-tailed P-value If it’s unlikely to observe such a small test statistic, i.e., if the P- value (0.0127) is smaller than α, reject the null hypothesis.

Two-tailed P-value If it’s unlikely to observe such an extreme test statistic, i.e., if the P-value (0.0254) is smaller than α, reject the null hypothesis.

Example: Right-tailed test Brinell hardness measurement of ductile iron subcritically annealed: One-Sample T: Brinell Test of mu = 170 vs mu > 170 Variable N Mean StDev SE Mean T P Brinell

Example: Right-tailed critical value

Example: Right-tailed P-value

Example: Left-tailed test Height of sunflower seedlings Test of mu = 15.7 vs mu < 15.7 Variable N Mean StDev SE Mean T P Sunflower

Example: Left-tailed critical value

Example: Left-tailed P-value

Example: Two-tailed test Thickness of spearmint gum Test of mu = 7.5 vs mu not = 7.5 Variable N Mean StDev SE Mean T P Gum

Example: Two-tailed critical value

Example: Two-tailed P-value