Estimation of Parameters (Interval Estimate)

Flashcards covering the concepts, definitions, and formulas related to interval estimation of parameters, including population mean and proportion. Remove the examples

Interval estimate of a parameter

An interval or a range of values used to estimate the parameter.

Confidence level of an interval estimate

The probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated; equivalent to (1 − 𝛼).

Confidence interval

A specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.

Confidence interval

𝑃𝑜𝑖𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 ± 𝑀𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟

Formula for a 100(1 − 𝛼)% confidence interval on μ when the population variance σ2 is known

ҧ𝑥− 𝑧 𝛼/2 𝜎/ 𝑛 ≤ 𝜇 ≤ ҧ𝑥+ 𝑧 𝛼/2 𝜎/ 𝑛

Best point estimate of the true average score

The sample mean.

Standard error of the point estimate (population standard deviation is 20, sample size is 100)

20/sqrt(100) = 2

Margin of error formula when the population standard deviation is known

z(α/2) * (𝜎/ sqrt(𝑛))

95% confidence interval estimate for the true average score in Chemistry when the sample mean is 78 and the population standard deviation is 20 (sample size of 100)

74.08 ≤ 𝜇 ≤ 81.92

Formula for a 100(1 − 𝛼)% confidence interval on 𝑝 (population proportion)

Ƹ𝑝 − 𝑧 𝛼/2 * sqrt((𝑝𝑞)/𝑛) ≤ 𝑝 ≤ Ƹ𝑝 + 𝑧 𝛼/2 * sqrt((𝑝𝑞)/𝑛)

Point estimate of the percentage of all people who prefer to own safer cars

The sample proportion.

90% confidence interval for the percentage of all people who will not mind paying a few thousand dollars more to have safer cars in a sample where 44% of 500 people said they would

0.4035 ≤ 𝑝 ≤ 0.4765

Formula for a 100(1 − α)% confidence interval on μ when the population variance σ2 is unknown

ҧ𝑥− 𝑡 𝛼/2,𝑛−1 𝑠/ 𝑛 ≤ 𝜇 ≤ ҧ𝑥+ 𝑡 𝛼/2,𝑛−1 𝑠/ 𝑛

Distribution to use when n < 30 and the population standard deviation is unknown

Student's t-distribution

95% confidence interval for the true average grade in Mathematics from sample data

81.40 ≤ 𝜇 ≤ 90.40

If the 95% confidence interval contains the average grade of 85, what does this imply?

YES, the average grade of 85 is within the interval and it goes to show that the mean of our Interval Estimation is located at 85.

95% confidence interval for the population mean travel time

Mean Travel Time: 9.137 ≤ 𝜇 ≤ 17.313

Margin of error formula when population standard deviation is unknown

𝒕𝛼 2,𝑛−1 * s/sqrt(n)

Formula for 100(1 − 𝛼)% confidence interval on 𝑝1 − 𝑝2 when estimating the difference of population proportions

Difference of Sample Proportions − 𝑧 α/2 * sqrt((Sample Proportion 1 * Sample Proportion Complement 1)/Sample Size 1) + ((Sample Proportion 2 * Sample Proportion Complement 2)/Sample Size 2) ≤ Population 1 − Population 2 ≤ Difference of Sample Proportions + 𝑧 α/2 * sqrt((Sample Proportion 1 * Sample Proportion Complement 1)/Sample Size 1) + ((Sample Proportion 2 * Sample Proportion Complement 2)/Sample Size 2)

Formula for 100(1 − α)% confidence interval on 𝝁𝟏 − 𝝁𝟐 when estimating the difference of population means with known variances

Difference of Sample Means − 𝑧 α/2 * sqrt((Variance 1/Sample Size 1) + (Variance 2/Sample Size 2)) ≤ Population 1 − Population 2 ≤ Difference of Sample Means + 𝑧 α/2 * sqrt((Variance 1/Sample Size 1) + (Variance 2/Sample Size 2))

Formula for 100(1 − α)% confidence interval on 𝝁𝟏 − 𝝁𝟐 when estimating the difference of population means with unknown (equal) variances

Difference of Sample Means − 𝑡 𝛼/2,𝑑𝑓 * 𝒔𝒑 * sqrt((1/𝒏𝟏) + (1/𝒏𝟐)) ≤ 𝜇1 − 𝜇2 ≤ Difference of Sample Means + 𝑡 𝛼/2,𝑑𝑓 * 𝒔𝒑 * sqrt((1/𝒏𝟏) + (1/𝒏𝟐))