Chapter 8 - Confidence intervals for one population mean

point estimate

of a parameter is the value of a statistic used to estimate the parameter

• sample mean is an estimate of population mean

• sample proportion is an estimate of population proportion

confidence interval (interval estimator)

a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter

• indicates the degree of confidence we have in the estimate of target parameter

• a measure of reliability or accuracy

confidence interval (Ci)

an interval of numbers obtained from a point estimate of a parameter

confidence level

the confidence we have that the parameter lies in the confidence interval (ie that the confidence interval contains the parameter

confidence interval estimate

the confidence level and confidence interval

margin of error

the endpoints of confidence interval, found by subtracting and adding 𝜎x̄ to the sample mean

• MOE is half the length of the confidence interval

endpoints

can express endpoints of the confidence interval as follows…

point estimate +- margin of error

written form of confidence level

1 - a

1 - a key

a = number between 0-1, the number that MUST be subtracted from 1 to get confidence level

ex. confidence level = 95%

then a = 1-0.95 = 0.05

z_a

z-score that has area a to its right under the standard normal curve

z_a/2

z score that has area a/2 to its right

• both sides of the normal curve, high and low end 

• used in confidence intervals

basis of confidence interval

• x is normally distributed w/ mean μ and standard deviation σ

• sample size of n

• variable x̄ is also normally distributed w/ mean μ and standard deviation   σ/√n

• use empirical rule to conclude approx. 95% of all samples of size n have means within 2xσ/√n (2 standard deviations) of the mean μ

steps for one mean z-interval procedure (σ known)

1. for confidence level of 1-a, find Z_a/2

Z_a/2 is found, n is sample size, and x̄ is computed from some data

2. confidence interval for μ is from x̄ - Z_a/2 * σ/√n to x̄ + Z_a/2 * σ/√n

3. interpret the confidence interval

Ci is exact for norm. populations (means true confidence level equals 1-a) and approx. correct for large samples from non normal populations (means true confidence level only approx. equal)

when to use the one mean z-interval procedure

• small samples

• moderate sample sizes

• large sample sizes

small samples

(less than 15) used when variable is norm. distributed or v close

moderate sample sizes

(15-30) used unless data contains outliers or variable is far from normal distribution

large samples

(30+) used without restriction (but check for outliers using box plots)

Table 11 / 8.3

Confidence level

90% a = 0.10 Za/2 = 1.645

95% a = 0.05 Za/2 = 1.960

99% a = 0.01 Za/2 = 2.575

margin of error for estimate of μ formula

Z_a/2 * σ/√n = E

• equals half the length of confidence interval

• can use MOE or length of confidence interval to measure accuracy of point estimate

MOE

indicates the accuracy of our confidence interval estimate (the accuracy with which a sample mean estimates the unknown population mean)

MOE sizes

• small MOE = good accuracy

• large MOE = poor accuracy

• size can be controlled by confidence level or sample sizes

decreasing confidence interval

decreases margin of error = improvement of accuracy

increasing sample size

decreases MOE = improvement of accuracy

sample size for estimating μ formula

n = (Z_a/2 * σ / E )² 

• round to nearest whole #

students t-statistic

has sampling distribution like normal distribution (z-statistic)

1. round shaped 

2. symmetric

3. mean of 0

just has more variables (wider)

t-distributions

• different t-distribution for each sample size

• identify particular t-distribution by its # of degrees of freedom (df)

• amount of variability in the sampling distribution depends on the sample size 

degrees of freedom formula

n-1

• number of values in a set of scores that are free to vary after certain restrictions are placed on data

studentized version of sample mean formula

t = x̄-μ / S√n

studentized version of sample mean key

x̄ = sample mean

μ = population mean (true average)

S = sample standard deviation (how spread the data is)

n = sample size 

t - t-score (how far your sample mean is from population)

basic properties of t-curves

prop 1. total area under t-curve = 1

prop 2. curve extends indefinitely in both directions

prop 3. curve is symmetric about 0

prop 4. as # of df becomes larger t-curves look increasingly like the standard normal curve

t table

t_a t-value having area (a) to its right under a t-curve

one mean t-interval procedure (𝜎 unknown)

1. for confidence level of 1-a, find t_a/2 with df = n-1 (n = sample size)

where t_a/2 is found and x̄ and s are computed from sample data

2. The confidence interval for μ is from  x̄ -  t_a/2 *S√n to  x̄ + t_a/2 *S√n

3. interpret the confidence level