Statistics 1 - Final

Describe the shape, center, and spread of the sampling distribution of the sample mean when the distribution is normal.

Shape: normal regardless of sample size

Center: mean (xbar) = mean of population

Spread: standard deviation(xbar) = standard deviation/root(n)

Describe the shape, center, and spread of the sampling distribution of the sample mean when the distribution is not normal.

Shape: as the sample size increases, the distribution of the sample mean becomes normal

Center: mean (xbar) = mean of population

Spread: standard deviation(xbar) = standard deviation/root(n)

Central limit theorem

Regardless of the shape of the underlying population, the sampling distribution of xbar becomes approximately normal as the sample size, n, increases (The distribution of the sample mean is normal when n >= 30)

Describe the shape, center, and spread of the sample proportion (p-hat)

Shape: as sample size increases, shape becomes normal provided np(1-p) >= 10

Center: mean of the distribution of the sample proportion = population proportion (p)

Spread: as the sample size increases, the standard deviation decreases

Confidence interval

a range of values that is likely to contain a population parameter, providing an estimate of its uncertainty

Level of confidence

the percentage of times a confidence interval will contain the true population parameter if the study were repeated many times

Margin of error

the range of uncertainty or variability around an estimate or measurement

Critical value

a threshold in statistics that helps determine if your results are statistically significant

T-distribution

number of standard deviations away from the middle of a sample mean when the population standard deviation is unknown and the sample size is small

What are the 6 rules of a t-distribution?

1. The t-distribution is different for different degrees of freedom

2. Centered around 0 and symmetric about 0

3. Area under the curve is 1. The area under the curve to the right of 0 equals the area under the curve to the left of 0, which equals ½.

4. As t increases or decreases without bound, the graph approaches, but never equals zero

5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further variability into the t-statistic

6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because, as the sample size n increases, the values of s get closer to the values of σ, by the Law of Large Numbers

t-interval

a statistical tool that estimates a range of likely values for an unknown population mean, especially when the sample size is small and the population's standard deviation is unknown

What is the criteria for a t-interval

1. the sample is random

2. the sample is small relative to the population size (n<=0.05N)

3. is normally distributed

4. does not have outliers

5.  sample size is <30

Chi-Squared test 

a test used to determine if there is a significant difference between observed and expected frequencies for categorical data

Criteria for a chi-squared test

1. Not symmetric

2. Depends on degrees of freedom

3. As the degree of freedom increases, the chi-square becomes nearly symmetric

4. Always positive

What are the 4 outcomes of hypothesis testing?

1. Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct.

2. Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct

3. Reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error

4. Do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a Type II error.

Null hypothesis (H0)

no change/effect/difference, assumed true until evidence indicates otherwise

Alternative hypothesis (H1)

alternative to the null hypothesis

What are the 3 tests for a null hypothesis?

1. Equal vs unequal hypothesis (two-tailed)

H0 = some value

H1 != some value

2. Equal vs less-than (left-tailed)

   H0 = some value

   H1<some value

3. Equal vs greater than (right-tailed)

H0 = some value

H1 > some value

Level of signficance (alpha)

the probability of making a Type I error

How do you calculate the standard deviation of a sample proportion

Use formula: standard deviation = root(p * (1-p)/n)

How do you calculate a point estimate with an upper and lower bound?

Point estimate = (lower bound + upper bound)/2

How do you calculate a confidence interval?

1. Calculate sample proportion (p-hat) = x/n

2. Determine critical value (z a/2) for given confidence level -> a/2 = (1-confidence level)/2 -> find z-score such that the area to the right is a/2

3. Calculate the margin of error: E = z(a/2) * root(p-hat x (1-p)/n)

4. Calculate lower bound -> lower bound = p-hat - E

5. Calculate the upper bound -> upper bound = p-hat + E

How do you calculate the level of confidence?

Level = (1-alpha) x 100%, where alpha is the error (or what is outside of that interval

How do you calculate the margin of error with upper/lower bounds?

Margin of error = (Upper bound - lower bound)/2

How do you calculate how large a sample needs to be?

1. Calculate p-hat: (p-hat) = x/n

2. Find Z(a/2) such that a/2 = (1-confidence level)/2

3. Calculate the margin of error (use either formula depending on what’s given)

4. Plug all values into the formula: n = (p-hat * (1 - p-hat))(z(a/2)/E)²

How do you calculate a T-distribution?

t = (x-bar - mean)/(s/root(n))

How do you construct a T-interval?

1. Determine degrees of freedom: df = n-1

2. Calculate a/2: a/2 = (1 - confidence interval)/2

3. Find t-value for df and a/2: use the table

4. Calculate margin of error: E = t-value x (s/root(n))

5. Calculate lower bound: x-bar - E

6. Calculate upper bound: x-bar + E 

How do you calculate the margin of error for a t-interval?

1. Find t(a/2) where a/2 = (1-confidence level)/2

2. Plug into the formula: E = t(a/2) x (s/root(n))

How do you calculate a chi-squared value?

X² = ((n-1)s²)/(standard deviation)²

How do you construct a confidence interval for (standard deviation)²?

1. Determine degree of freedom: df = n-1

2. Find lower tail chi-squared critical value: a/2 = (1-confidence level)/2

3. Find upper tail chi-squared critical value: 1 - (a/2)

4. Find chi-squared critical values: use the table for lower and upper tails

5. Calculate the lower bound of the confidence interval: ((n-1)s²)/(X(a/2)²

6. Calculate the upper bound of the confidence interval: ((n-1)s²)/(X(1 - a/2)²

7. If asking for standard deviation, square root results