AP Stats Chapter 10

Front

What is the t-distribution and when is it used?

Back

The t-distribution is a probability distribution used in statistics that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used when the sample size is small (typically n < 30) and the population standard deviation is unknown.

Front

Why do we use the t-distribution instead of the z-distribution for small samples?

Back

We use the t-distribution instead of the z-distribution for small samples because the t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.

Front

What is a confidence interval?

Back

A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. It is expressed with a confidence level, such as 90% or 95%.

Front

How do you calculate a confidence interval for the difference between two proportions?

Back

To calculate a confidence interval for the difference between two proportions, use the formula: (p1 - p2) ± Z * sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)], where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and Z is the Z-value corresponding to the desired confidence level.

Front

What are the conditions for performing a two-sample t-test?

Back

The conditions for performing a two-sample t-test are: 1) The samples must be independent. 2) The data in each group should be approximately normally distributed, especially for small sample sizes. 3) The variances of the two populations should be equal (or approximately equal).

Front

What does it mean for a distribution to be normal?

Back

A distribution is considered normal if it is symmetric about the mean, with a bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions.

Front

What is the significance of degrees of freedom in the t-distribution?

Back

Degrees of freedom in the t-distribution refer to the number of independent values that can vary in an analysis without violating any constraints. In a t-test, degrees of freedom are typically calculated as the total number of observations minus the number of groups.

Front

What is the relationship between sample size and the t-distribution?

Back

As the sample size increases, the t-distribution approaches the normal distribution. This is because larger samples provide more information about the population, reducing the uncertainty in the estimate of the population standard deviation.

Front

What is a two-sample t-test used for?

Back

A two-sample t-test is used to determine if there is a statistically significant difference between the means of two independent groups.

Front

What is the formula for calculating the t-statistic?

Back

The formula for calculating the t-statistic is: t = (X̄1 - X̄2) / sqrt[(s1²/n1) + (s2²/n2)], where X̄1 and X̄2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.