Unit 4: Inference for Quantitative Data: Means

15 terms. Say what each one means, then open the card to check.

• t distribution

  A family of symmetric, bell-shaped curves centered at 0 with heavier tails than the standard Normal curve. Which curve you use depends on the degrees of freedom.

  Example. With 9 degrees of freedom, the middle 95% of the t curve runs from -2.262 to 2.262, wider than -1.96 to 1.96 for z.

• Degrees of freedom (df)

  The number that picks which t curve to use. For a one-sample t procedure it is n - 1. As df grows, the t curve gets closer to the standard Normal curve.

  Example. A random sample of 25 battery lifetimes gives df = 24 for a t interval for the mean lifetime.

• Standard error of the mean

  The estimate $\frac{s}{\sqrt{n}}$ of how far a sample mean typically falls from the population mean. It uses the sample standard deviation s because the population one is unknown.

  Example. For a sample of 36 wait times with s = 12 minutes, the standard error is 12 divided by 6, which is 2 minutes.

• Critical value t*

  The t value that cuts off the middle C% of the t curve for the right df. It sets how many standard errors wide each side of the interval is.

  Example. For 95% confidence with df = 24, t* is about 2.064, so the margin of error is 2.064 standard errors.

• Margin of error

  The distance from the sample mean to each end of a confidence interval: the critical value times the standard error. It is half the width of the interval.

  Example. An interval of 41.0 to 47.0 minutes has its center at 44.0 and a margin of error of 3.0 minutes.

• Confidence level

  The long-run share of intervals, built the same way from many random samples, that would capture the true parameter. It describes the method, not one interval.

  Example. At 90% confidence, about 90 of every 100 intervals from repeated random samples would contain the true mean.

• Normal/Large Sample condition

  For t procedures, the population is roughly Normal or n is at least 30. With a small sample, check a graph of the data for strong skew or outliers.

  Example. With only 10 values, a dotplot that is roughly symmetric with no outliers makes a t interval reasonable.

• 10% condition

  When sampling without replacement, the sample should be no more than 10% of the population, so the observations can be treated as independent.

  Example. A sample of 40 students from a school of 900 passes, since 40 is less than 90.

• Null hypothesis for a mean

  The claim of no effect or no change that a test starts by assuming, written about the population mean, as in $H_0: \mu = \mu_0$.

  Example. To check a label that says 500 grams, the null hypothesis is that the mean weight of all boxes is 500 grams.

• One-sample t statistic

  $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$: how many standard errors the sample mean is from the value in the null hypothesis.

  Example. A sample mean of 496.8 grams with a standard error of 1.6 grams gives t = -2.00 when the null value is 500 grams.

• P-value

  Assuming the null hypothesis is true, the probability of a test statistic at least as extreme as the one observed, in the direction of the alternative.

  Example. A p-value of 0.03 means that if the true mean were the null value, only about 3% of samples would look this extreme.

• Type II error

  Failing to reject the null hypothesis when the alternative is actually true. Its probability falls as the sample size or the true effect gets larger.

  Example. A machine really underfills bottles, but the test finds no convincing evidence of underfilling.

• Paired data

  Two measurements that come in matched pairs, such as one person measured twice or twins split between treatments. You analyze the differences within each pair.

  Example. Each of 15 runners times a sprint in two shoe models, so each runner gives one difference.

• Mean difference $\mu_d$

  The population mean of the paired differences. Paired t procedures are one-sample t procedures on the differences, with df = number of pairs minus 1.

  Example. For 12 typists, the differences new keyboard minus old keyboard have a mean that estimates $\mu_d$, with df = 11.

• Two-sample t procedures

  Inference for $\mu_1 - \mu_2$ from two independent groups, using the standard error $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ and df from technology.

  Example. Comparing the mean battery life of 30 phones of one model with 35 phones of another uses two-sample t procedures.