STA ch6-8

Interpretation of the CI:

is that at a XX% CI, any value of the parameter within that interval is plausible; in the long run, XX% of the computed CI's will contain mu.

P-value is often referred to as the observed significance level (OSL):

it is the smallest value of a for which 𝐻0 can be rejected.

Estimator is unbiased if

its probability (sampling distribution) is centered at the true value of the parameter for every possible value of the parameter

Case 2:

large sample tests.

T distribution

more spread out than normal distribution

Case 3:

normal population distribution with small n.

Case 1:

normal population with known standard deviation

Which hypothesis is the hypothesis that is initially assumed to be true:

null hypothesis

select a significance level α, then

reject H0 if P-value ≤ α do not reject H0 if P-value > α

Which of the parameters are used to calculate the confidence interval for the population mean (large sample size, any probability distribution)?

sample standard deviation, sample size, sample mean

The higher the desired degree of confidence:

the wider the resulting interval will be.

Approach should be to state the largest value of α that can be tolerated and the resulting value of α is often referred to as the significance level of the test

true

Confidence intervals provide an interval of plausible values for the parameter being estimated

true

For binomial random variable X with parameters n and p, the sample proportion is an unbiased estimator of p.

true

One sample t test is used when we have a normal population distribution and small n.

true

When selecting from unbiased estimators, an estimator with small bias may be preferred over the minimum variance unbiased estimator when the variance of the estimator with bias is less than for the MVUE.

true

θ0 is the null value and appears in both the null and alternative hypothesis

true

When selecting among different estimators, select one that is:

unbiased

To achieve a different CI than 95%, one would have to change:

α

Narrower interval:

is more precise but the CI would be lower and so less reliable.

p-value

-calculated assuming null hypothesis is true -a probability (b/w 0 and 1) -not probability that H0 is true, nor an error probability

Statistical hypothesis is a claim or assertion either about the value of:

A. A single parameter (population characteristic or characteristic of a probability distribution) B. About the values of several parameters C. About the form of an entire probability distribution ****D. Any of the above

For Ho: θ = θ0, the alternative hypothesis will look like:

A.Ha: θ > θ0 (in which case the implicit null hypothesis is θ ≤ θ0) B. Ha: θ < θ0 (in which case the implicit null hypothesis is θ ≥ θ0) C. Ha: θ ≠ θ0 ****D. Any of the above three

What is a parameter in point estimation?

Any statistic of the population we are interested in estimating

Bias of the estimator is the ________ between the expected value of the estimator and true value of the parameter

Difference

What is the objective of point estimation?

Estimate the true value of a parameter

• Case II:

Large sample CI with standardized variable Z has approximately a standard normal distribution as long as the sample size Is sufficiently .

Case III:

Normal distribution with small n When n is small, the sample standard deviation will likely not be close to the population standard deviation so the variability in Z is from both the numerator (X bar-sample mean) and denominator (S-sample standard deviation).

Case I for population mean:

Normal distribution, σ is known

For a random sample from a distribution with mean µ and standard deviation σ, the unbiased estimator of population mean is:

Sample mean (could b sample median if we had a fairly symmetrical distribution)

For a binomial RV with parameters n and p, the unbiased estimator of p is:

Sample proportion

For a random sample from a distribution with mean µ and standard deviation σ, the unbiased estimator of population standard deviation is:

Sample standard deviation

The notation ϴ � refers to both the estimator of ϴ and the point estimate from a given sample

True

Rejecting the null hypothesis H0 when it is true (σ) is:

Type I error

Which type of error is more serious:

Type I error

confidence level (reliability) is __________________ to precision

inversely related, higher CI means a wider interval so we have more reliability but less precision because the endpoints are further apart

The higher the desired degree of confidence, the resulting interval will be:

Wider

Higher CI means:

Wider interval, more reliable, less precise

Case I - Standardized variable is:

Z = 𝑋−µ /( σ/root(n))

Minimum variance unbiased estimator (MVUE) -

among all estimators of a parameter that are unbiased, select one that has minimum value