Mod 8 Flashcards
(21 cards)
How do you estimate population mean μ?
Estimate population mean μ using sample mean x-bar (x-bar is typically different from μ)
Notation:
μ = mean for population
𝞼 = standard deviation for population
What is the sampling distribution of x-bar?
Sampling distribution of x-bar ~ AN (μ, 𝞼 /√ n)
How is the value of the standard deviation is related to sample size.
- Sample size increases = smaller standard deviation, gets closer to μ
- If you get a sample size 4x larger, you cut the standard deviation in half
When is the sampling distribution of x-bar is approximately normal?
1) If the population distribution of μ is normal, the sampling distribution of x-bar is also normal for any sample size n
2) If the population distribution of μ isn’t normal, the sampling distribution of x-bar is approximated to have an normal curve when n ≥ 30 (central limit theorem)
How do you solve problems trying to estimate population mean/or estimating population mean using sample mean?
- If population only = AN (μ, 𝞼) → normcdf (lower bound, upper bound, μ, 𝞼)
- If with a sample size = AN (μ, 𝞼/√ n) → normcdf (lower bound, upper bound, μ, 𝞼/√ n)
How do you compute margin of error when sample mean (x-bar) is used to estimate population mean (μ)?
(t*)(s/ √ n)
What happens to standard error if sample size increases?
Sample size increases = s estimates 𝞼 more accurately or s becomes very close to 𝞼
If you use standard error of x-bar instead of standard deviation, what is the distribution?
What happens if df increases?
When we replace 𝞼 with s, the value does not have a standard normal distribution. The added variability of s has a t-distribution, which is bell-shaped with wider tails and is characterized by df (n-1)
As df increases, spread of the t-distribution decreases
How do you compute the standard error of x-bar?
s/√ n
What is the formula for CI when 𝞼 known?
x-bar +/- (z*)(𝞼/√ n)
What is the formula for CI when 𝞼 unknown?
X-bar +/- (t*)(s/√ n)
- Multiplier is a t* value from t-distribution with n-1 degrees of freedom
What happens to margin of error and CI width if confidence level increases?
- Increase confidence level = increase the margin of error
- Increase confidence level = increase CI width
What happens to margin of error, standard error, and CI width if sample size increases?
- Increase sample size (n) = decrease the margin of error
- Increase sample size (n) = decrease standard error, CI width
How do you interpret confidence interval?
If we construct many confidence intervals using this interval, 95% of them should cover/contain the true population mean
How to find critical values from the t-distribution?
To find 95% CI based on n = 10 observations
Get t-dist (the left-tail area, df (n-1)) = t-dist(97.5, 9)
How do you calculate sample size for a desired margin of error to estimate a population mean?
n = (z* 𝞼/M)^2
When 𝞼 is unknown, we estimate it with range/4
Translate a research question or claim about a population mean into null and alternative hypotheses
H0: μ = μo
Ha: μ < μo, μ > μo, μ ≠ μo
Use all steps to conduct a t test of hypotheses about a population mean
Hypotheses
Conditions
- The population of measurements is approximately normal and random sampling is used to obtain data
- The population of measurements is nonnormal, but a large random sample is measured (n ≥ 30)
Test statistic: x-bar - μ/s/√ n
- Test statistic t will be large when x-bar is far from μ
- Test statistic t will be small when x-bar is close to μ
P-value: p-value will be small when is far from μo, and large when is close to μo
Ha: μ > μo → t.dist (x, df, 1)
Ha: μ < μo → t.dist (x, df, 1)
Ha: μ ≠ μo → t.dist (x, df, 1)
Conclusion in context
- Statistically significant (reject H0) if p-value ≤ alpha
- Not statistically significant (fail to reject H0) if p-value ≥ alpha
What is the distribution of the test statistic when the null is true?
When the null is true, test statistic follows a normal distribution or t-distribution
What is type I error? What is type II error?
Type I error: rejecting the null when the null is true
Type II error: fail to reject the null when the null is false
What is power? What happens if sample size (n) increases? When else does power increase?
Power: probability that we reject the null when a value in the alternative is the truth (when we should reject the null)
- Sample size doesn’t affect the probability of type I error
- As sample size increases, the probability of type II error decreases
- sample size increases, power increases
- When the difference between true population value and null hypothesis value increases, power increases
