Chapter 8

Question 1

Confidence Interval for the Mean (σ known)

Answer 1

This is essentially a “known probability” (which is for a population) but for a sample.

i.e. you’re substituting x̄ for µ

Question 2

Confidence Interval for the Mean (σ unknown)

Answer 2

where

t_α/2 is the critical value corresponding to an upper-tail probability of α/2 from the t distribution with n - 1 degrees of freedom

Question 3

Level of Confidence

Answer 3

is symbolized by (1 - α) * 100%, where

α is the proportion in the tails of the distribution that is outside the confidence interval

Question 4

Student’s t distribution

Answer 4

t = x̄ - µ ÷ (S / sqrt-n)

If the variable X is normally distributed, the the following statistic has a t distribution with n - 1 degrees of freedom

Question 5

Standard Confidence Intervals

90%

95%

99%

Answer 5

90% = .95 & .05

95% = .975 & .025

99% = .995 & .005

Question 6

Confidence Interval Estimate for the Proportion

Answer 6

where

p = sample proportion = X/n

π = population proportion

Z_α/2 = critical value from the standardized normal distribution

n = sample size

Question 7

Sample Size Determination for the Mean

Answer 7

To compute the Sample Size, you must know three quantities:

1. The desired confidence level, which determines the value of Zα/2, the critical value from the standardized normal distribution
2. The acceptable sampling error, e
3. The standard deviation, σ

Question 8

Sampling error e

Question 9

Backing into σ for confidence intervals

Answer 9

if you assume a normal distribution, the range is approximately equal to 6σ (± 3σ) so that you estimate σ as the range divided by 6

Question 10

Sample Size Determination for the Proportion

Answer 10

To determine the sample size needed for estimating the population mean

Developing the sample size for a confidence interval for the mean, the sampling error is defined by

• e* = Z_α/2*σ/sqrt-n
• When estimating a proportion, you replace* σ with sqrt-π(1-π), thus the sampling error is
• e* = Z_α/2*sqrt[(π(1-π)/n]

which leads to…

Question 11

Values for π and π(1-π)

Answer 11

When π = 0.9, then π(1-π) = (0.9)(0.1) = 0.09

When π = 0.7, then π(1-π) = (0.7)(0.3) = 0.21

When π = 0.5, then π(1-π) = (0.5)(0.5) = 0.25

When π = 0.3, then π(1-π) = (0.3)(0.7) = 0.21

When π = 0.1, then π(1-π) = (0.1)(0.9) = 0.09

Question 12

Confidence Interval Estimate of the Mean

Answer 12

(1- α)*100%

99% confidence interval would be

(1 - α)*100% = 99% > (1 - α) = .99 > 1 - .01 = α > α = .01

Find 1 - α/2 to get the area under the curve for 99%

then find Z_α/2 = Z_0.005 >

Question 13

Degrees of Freedom

Answer 13

n - 1

Question 14

Standard Error

Answer 14

σ / sqrt-n