AP Stats Unit 7

df =

n - 1

t* can be found with

invT(area to the LEFT, df)

Conditions for calculating a confidence interval for a population mean is…

Random: The data comes from a random sample from the population of interest

10%: n < 0.1(N)

Normal/LS:

• Population dist. is approximately normal

• n≥30

• n<30 but the dotplot/boxplot of data shows no strong skew out outlier

Standard error (SE) of the sample mean x_bar =

S_{x_bar} = S_x / sqrt(n)

Formula for the confidence interval for a population mean is…

x_bar +- (t*)(S_{x_bar})

A (???) is confidence interval used to estimate a population mean μ when the population standard deviation σ is unknown.

“one sample t interval for μ”

Calc Command: TInterval

Conditions for calculating a confidence interval for a population mean difference

n_diff = # of differences (# of paired pieces of information)

Random: Paired data come from a random sample from the population of interest OR from a randomized experiment

10%: When SAMPLING w/o replacement, n_diff < 0.1(N_diff)

Normal/LS:

• Population dist. for n_diff is approximately normal

• n_diff≥30

• n_diff<30 but the dotplot/boxplot of data shows no strong skew out outlier

Formula for calculating the confidence interval for a population mean difference…

x_bar_diff +- (t*)(S_diff / sqrt(n_diff))

*Use the mean and stdev of the differences

A (???) is a confidence interval used with paired data to estimate a population mean difference.

One-sample t interval for mean difference

Calc Command: TInterval (here the μ is simple replaced with μ_diff)

Conditions for performing a significance test about a population mean…

Random: The data comes from a random sample from the population of interest

10%: n < 0.1(N)

Normal/LS:

• Population dist. is approximately normal

• n≥30

• n<30 but the dotplot/boxplot of data shows no strong skew out outlier

Calculating the standardized test statistic and p-value in a test about a population mean

H_0: μ = μ_0

t = (x_bar - μ_0) / (S_x / sqrt(n))

p-value = tcdf (NOT normalcdf)

A (???) is a significance test of the null hypothesis that a population mean μ is equal to a specified value, when the population standard deviation σ is unknown.

one-sample t test for a mean

Calc Command: T-Test

Conditions for performing a significance test about a population mean difference

n_diff = # of differences (# of paired pieces of information)

Random: Paired data come from a random sample from the population of interest OR from a randomized experiment

10%: When SAMPLING w/o replacement, n_diff < 0.1(N_diff)

Normal/LS:

• Population dist. for n_diff is approximately normal

• n_diff≥30

• n_diff<30 but the dotplot/boxplot of data shows no strong skew out outlier

Calculating the standardized test statistic and p-value in a test about a population mean difference

H_0: mu_diff = 0
t = (x_bar_diff - 0) / (S_diff / sqrt(n_diff)) w/ df = n_diff - 1

p-value = tcdf (NOT normalcdf)

A (???) is a significance test of the null hypothesis that a population mean difference is equal to a specified value, usually 0.

One sample t test for μ_diff

Calc Command: T-Test (here the μ is simple replaced with μ_diff)

Conditions for constructing a confidence Interval for a Difference between two Population Means

Random: The data comes from two independent random samples OR from two groups in a randomized experiment

10%: When sampling w/o replacement n1 < 0.1(N1) and n2 < 0.1(N2)

Normal/LS (for each n1 AND n2):

• Population dist. for n1 (and n2) is approximately normal

• n1 (and n2) ≥30

• n1 (and n2) <30 but the dotplot/boxplot of data shows no strong skew out outlier

Calculating a confidence interval for a difference between two population means

(x_bar_1 - x_bar_2) +- (t*)(sqrt(S_1 ² / n1 + S_2 ² / n2))

df = smaller of the n1 - 1 and n2 - 1

A (???) is a confidence interval used to estimate a difference in the means of two populations or treatments with unknown standard deviations.

two-sample t interval for a difference in means

Calc command: 2-SampTInt

Conditions for performing a significance test about a difference between two population means

Random: The data comes from two independent random samples OR from two groups in a randomized experiment

10%: When sampling w/o replacement n1 < 0.1(N1) and n2 < 0.1(N2)

Normal/LS (for each n1 AND n2):

• Population dist. for n1 (and n2) is approximately normal

• n1 (and n2) ≥30

• n1 (and n2) <30 but the dotplot/boxplot of data shows no strong skew out outlier

Calculating the standardized test statistic and p-value in a test for a difference between two population means

H_0: u1 - u2 = 0
([x_bar_1 - x_bar_2] - 0) / [sqrt(S_1 ² / n1 + S_2 ² / n2))]

df = smaller of n1 - 1 and n2 - 1

A (???) is a significance test of the null hypothesis that the difference in means of two populations or treatments is equal to a specified value (usually 0), when both population standard deviations are unknown.

two-sample t test for a difference in means

Calc Command: 2-SampTTest

Paired or Not Paired (μ_diff OR μ1 - μ2)?

1. If the data sets have different sample sizes, then NOT μ_diff

2. Questions about μ_diff typically have the phrase “mean difference” while μ1 - μ2 typically use the phrase “difference in means”