Statistics Chapter 7
point estimate
A ________ ________ is a specific value of a sample statistic from one random sample.
100/2500 = .04 < .05. NO FPC E(xbar) = μ = $51,800 σ(xbar) = σ/√n = $400
Calculate E(xbar) and σ(xbar) for a sample with 100 managers. Recall that the population mean and standard deviation for all 2500 managers are μ = $51,800 and σ = $4,000 respectively.
200/2500 = .08 > .05. YES FPC E(xbar) = μ = $51,800 σ(xbar) = (σ/√n)(√((N - n)/(N - 1)))= $271.35
Calculate E(xbar) and σ(xbar) for a sample with 200 managers. Recall that the population mean and standard deviation for all 2500 managers are μ = $51,800 and σ = $4,000 respectively.
(50)(.6) = 30 > 5 (50)(.4) = 200 > 5 50/2500 = .02 < .05 NO FPC σ(pbar) = sqrt(p(1-p)/n) σ(pbar) = sqrt((.6)(.4)/50) = .069
Compute the expected value standard error and distribution shape for the sample proportion (pbar) computed using a random sample of 50 EAI managers. Assume that the population proportion for all 2500 managers who completed a certain training program is p = .6.
P(xbar > 51800+400) = P(xbar > 52300) =normalcdf(lower, upper, sample mean, sample SD) =normalcdf(52300, E99, 51800, 400) =.1056
For a sample where the population mean and standard deviation for all 2500 managers are μ = $51,800 and σ = $4,000 respectively, what is the probability that the sample mean (xbar) computed using a random sample of 100 EAI managers will be more than $500 above the population mean?
P(51300 < xbar < 52300) =normalcdf(lower, upper, sample mean, sample SD) =normalcdf(51300, 52300, 51800, 400) =.7887
For a sample where the population mean and standard deviation for all 2500 managers are μ = $51,800 and σ = $4,000 respectively, what is the probability that the sample mean (xbar) computed using a random sample of 100 EAI managers will be within $500 of the population mean?
larger
In terms of consistency, the point estimator with the (larger / smaller) sample size is preferred because it provides estimates closer to the population parameter.
Smaller
In terms of efficiency, the point estimator with the (larger / smaller) standard error is preferred because it provides estimates closer to the population parameter.
E(pbar) = .44 σ(pbar) = .0496
In the United States, 44% of the population has type O blood. Suppose a random sample of 100 people is taken. What are the expected mean and standard deviation for the proportion of people having type O blood in our sample of 100?
=normalcdf(.42, E99, .44, .0496) = .6566
In the United States, 44% of the population has type O blood. Suppose a random sample of 100 people is taken. What is the probability that the sample proportion with type O blood will be greater than 0.42?
x = the number of elements in the sample that possess the characteristic of interest n = sample size
In the equation (pbar = x / n), what do x and n represent?
σ = 4000 / √30 σ = $730.30
Suppose the EAI company took MANY samples of size n = 30 managers from the population of N = 2,500 and calculated xbar, σ, and pbar for each sample. The population standard deviation for salary was $4000. Calculate the standard deviation of one of these samples (σ of xbar).
σ(pbar) = .0693 =normalcdf(0, .5, .6, .0693)
Suppose the head of HR tasks her assistant to take a random sample of 50 EAI employees and determine the proportion of the sample that has completed the training program. The assistant found that only 50% of the sample had completed the manager training. What is the probability that the sample proportion of managers who completed the training program, (pbar), computed using a random sample of 50 EAI managers will be less than 0.50? Recall that N = 2500 and p = .6.
False, (greater relative efficiency) = (smaller standard error)
T/F: (greater relative efficiency) = (larger standard error)
True, but it doesn't always have to be
T/F: A point estimate can be equal to the population parameter.
True
T/F: E(xbar) = μ and E(pbar) = p.
False, it applies to both
T/F: The FPC applies to xbar but not pbar.
False, sample size is irrelevant in calculating expected value among unbiased point estimators.
T/F: Unbiased point estimators from larger samples typically have more accurate expected values.
Central Limit Theorem 30
The ________ ________ ________ states that a sampling distribution. xbar will be approximately normally distributed if the sample size, n, is greater than or equal to ________.
xbar
The mean of a sample statistic is represented by what symbol?
finite population correction (FPC) FPC is necessary when n/N is greater than .05 (5%)
The pictured equation is known as the ________ ________ ________ factor, and is used when the sample of a finite population is (less / greater) than 5% of the total population.
both
The pictured equation is used to calculate the mean of a (population parameter / sample statistic / both)
sample statistic
The pictured equation is used to calculate the standard deviation of a (population parameter / sample statistic / both)
Infinite
The pictured formula is used to calculate the standard deviation for a (finite / infinite) population.
Infinite
The pictured formula is used to calculate the standard error for a (finite / infinite) population.
both
The pictured scenario is an example of how to calculate a proportion for a (population parameter / sample statistic / both)
sampling distributions
The probability distributions of point estimators are referred to as ________ ________.
pbar
The proportion of a sample statistic is represented by what symbol?
=normalcdf(166, 186, 176, 30) = .2611
The serum cholesterol levels of a population of 18-year-olds follows a normal distribution with mean 176 mg/dLi and a standard deviation of 30 mg/dLi. Estimate the chance that a randomly selected 18-year-old has a serum cholesterol level between 166 and 186 mg/dLi.
E(xbar) = 176 σ(xbar) = 30/√9 = 10 =normalcdf(166, 186, 176, 10) = .6827
The serum cholesterol levels of a population of 18-year-olds follows a normal distribution with mean 176 mg/dLi and a standard deviation of 30 mg/dLi. Suppose that we were to choose nine 18-year-olds at random from the population. What is the probability that the mean serum cholesterol level of the nine teenagers is between 166 and 186 mg/dLi?
p(7/9) n = 9 p = .2611 =binompdf(9, .2611, 7) = .0016
The serum cholesterol levels of a population of 18-year-olds follows a normal distribution with mean 176 mg/dLi and a standard deviation of 30 mg/dLi. What is the probability that seven of the nine 18-year olds had a serum cholesterol level between 166 and 186 mg/dLi?
σ(x/pbar)
The standard error of a sample statistic is represented by what symbol?
An unbiased estimator's expected value is equal to the population parameter it is estimating. A point estimator is said to be biased if E(𝜃*) ≠ 𝜃
What is an unbiased estimator?
A point estimator is the mean, standard deviation or number of successes for a sample, whereas a population parameter represents the expected value of the mean, standard error, and expected value of the number of successes for a population.
What is the difference between a point estimator and a population parameter?
Standard deviation is calculated among an entire population whereas standard error is calculated for a sample. SD = σ SE = σ(xbar)
What is the difference between standard deviation and standard error?
σ(pbar) = .0693 =normalcdf(.55, .65, .6, .0693)
What is the probability that the sample proportion of managers who completed a training program, (pbar), computed using a random sample of 50 EAI managers will be within 0.05 of the population proportion p? Recall that N = 2500 and p = .6.
The sample is from a population that is normally distributed. n ≥ 30
What two conditions could be met for a sampling distribution of xbar to have a normal distribution?
The sample size is sufficiently large such that np ≥ 5 and n(1 - p) ≥ 5. The sample size n is less than 5% of the population size N.
What two conditions must be met in order for the standard error of a sample proportion (pbar) to be calculated WIITHOUT the FPC?
Unbiased
Xbar is a(n) (biased / unbiased) estimator for μ.
