Chpt 8 - Estimation of the Mean and Proportion Flashcards
What are inferential statistics?
Statistical methods that draw and measure the reliability of conclusions about a population based on information obtained from a sample of the population
Describe the relationship between point estimation, a point estimator and a point
Point estimation is the statistical process of finding an approximate value of some parameter.
The point estimator of the population mean (parameter) is the sample mean (X̄).
The point estimate is the product of the process. So the point estimate of the population mean is a value of the mean of a sample (x̄).
What is a confidence interval?
It is an estimated range of values (obtained based on a sample) which includes the population mean (μ) with certain probability
What is the half length of a confidence interval for the population mean (μ) denoted as?
c
What are the upper and lower limits of the confidence interval?
lower bound: x̄-c
higher bound: x̄+c
What is a confidence level?
Quantifies the level of confidence that the population mean lies in the interval
Basically, its the percentage of the confidence intervals that contain the population mean
Can we tell if the population mean is in the confidence interval, even if the confidence level is really high, say 99%?
No, it’s impossible
What is the equation for a confidence level?
x̄ +/- (Z α/2) x (σ/√n)
For the equation for the confidence interval, what is x̄?
The sample mean of a sample
For the equation for the confidence interval, what is
Zα/2?
The z-score whose right area under the standard normal density curve is α/2
For the equation for the confidence interval, what is σ?
The population standard deviation
The volumes of children’s Tylenol are expected to have a mean 100mL, but the exact volume varies from bottle to bottle.
A random sample of 50 bottles are picked and the exact volume of each bottle is measured.
Suppose that the average volumes is 98 mL and the population standard deviation is 2mL.
Identify the sample size, sample mean, and the distribution of X̄?
n = 50
x̄ = 98
Because n>30, the distribution is approximately normal
The volumes of children’s Tylenol are expected to have a mean 100mL, but the exact volume varies from bottle to bottle.
A random sample of 50 bottles are picked and the exact volume of each bottle is measured.
Suppose that the average volumes is 98 mL and the population standard deviation is 2mL.
How would we determine the lower and upper bounds of a 90% confidence interval?
Determine important values:
n = 50
x̄ = 98
σ = 2
Determine the Zα/2 value
(1 - α) 100% = 90%
1 - α = 0.9
α = 0.1
α/2 = 0.05
This is the right area, so we find the left area, 1-0.05 = 0.95
Then we go to table 2, find 0.95 in the middle values to find the z-score along the sides (1.645)
Determine σ/√n value
σ/√n = 2/√50 = 0.28284
Determine (Z α/2) x (σ/√n)
1.645 x 0.28284 = 0.4653
Determine upper limit:
x̄ + (Z α/2) x (σ/√n)
98 + 0.4653 = 98.4653
Determine lower limit:
x̄ - (Z α/2) x (σ/√n)
98 + 0.4653 = 97.5347
A 95% confidence interval is (97.45, 98.55).
What does this mean?
We are 95% confident that the interval (97.45, 98.55) contains the population mean
What is the value of c in a confidence interval?
(Z α/2) x (σ/√n)