Inferential Statistics Weeks 1 and 2

One-Sided Test and CI and HT

CL = 1 - (2 * α)

p̂

Sample proportion.

Power of a Test

1 - β. The probability of correctly rejecting the null hypothesis H(o).

Conditions for the CLT

1. Independence: Sampled observations must be independent. - Random sample/assignment - If sampling, without replacement, n < 10% of population. 2. Sample Size/Skew: Either the population distribution is normal, or if the population distribution is skewed, the sample size is large (rule of thumb n > 30).

Conditions for the Confidence Interval

1. Independence: Sampled observations must be independent. - Random sample/assignment - If sampling, without replacement, n < 10% of population. 2. Sample Size/Skew: Either the population distribution is normal, or if the population distribution is skewed, the sample size is large (rule of thumb n > 30).

Hypothesis Testing for a Single Mean

1. Set the hypothesis. 2. Calculate the point estimate. 3. Check conditions (from CLT): a. Independence b. Sample Size/Skew 4. Draw sampling distribution, shade p-value, calculate test statistic. 5. Make a decision, and interpret it in context of the research question.

Common Misconceptions about Confidence Intervals

1. The confidence level of a confidence interval is the probability that the true population parameter is in the confidence interval you construct for a single sample. Truth: The confidence level is equal to the proportion of random samples that result in confidence intervals that contain the true pop. parameter. 2. A narrower confidence interval is always better. Truth: This is incorrect since the width is a function of both the confidence level and the standard error. 3. A wider interval means less confidence. Truth: This is incorrect since it is possible to make very precise statements with very little confidence.

68, 95, 99.7 Rule for Confidence Intervals

68% of the pop. will be within one standard error of the mean, two SEs for 95%, and three SEs for 99.7%

Confidence Interval

A plausible range of values for the population parameter. More formally, can be computed as the sample mean plus/minus a margin of error (critical value corresponding to the middle XX% of the normal distribution times the standard error of the sampling distribution).

Alternative Hypothesis

A proposed deviation from the status quo.

Test Statistic

A value, determined from sample information, used to determine whether to reject the null hypothesis.

Z score (Critical Value)

A z-score (aka, a standard score) indicates how many standard deviations an element is from the mean. A z-score can be calculated from the following formula.

Unbiased Estimator

Does not naturally over or underestimate the parameter. It provides a "good" estimate.

Two-Sided Test and CI and HT

CL = 1 - α

What drawbacks are associated with using a wider interval?

CL, width, and accuracy are inversely proportional to precision.

Is is okay to use the median to estimate the mean since both are a measure of the center of a distribution. T/F?

FALSE. It is not okay to use the median to estimate the mean. Depending on the shape of the distribution, these may be two very different measures of center.

It is okay to use the standard deviation to estimate the IQR since both are measures of variability. T/F?

FALSE. It is not okay to use the standard deviation to estimate the IQR. Depending on the shape of the distribution, these may be two very different measures of spread.

Type 2 Error

Failing to reject the null hypothesis H(o) when the alternative H(A) is true.

In one-sided tests, the significance level must also be considered on both sides, even if dealing with only one tail.

For example, if α = 0.05 there will be a 90% confidence interval.

Sample Distribution

Frequency distribution of all the elements of an individual sample.

Confidence Interval Formula

If n has a decimal, usually round up.

10% Condition

If sampling without replacement n < 10% of the pop. because observations within the pop. are usually not independent. Large samples are good, but keep it proportional to the entirety of the population.

How to have high precision and with high accuracy?

Increases sample size.

Two-Sided Tests

Instead of looking for a divergence from the null in a specific direction, we are interested in a divergence in any direction.

Sample Size / Skew Condition

Larger samples alter the shape of the sampling distribution to better resemble a normal distribution. The more the skew, the higher the sample size you need for the CLT.

Type 1 Error Rate Formula

P(Type 1 Error I H(o) true) = α

σ

Population standard deviation.

σ^2

Population variance.

Null Hypothesis

Reinforcement of the status quo.

Type 1 Error

Rejecting the null hypothesis H(o) when the null hypothesis H(o) is true.

α (alpha)

Represents the significance level, which is usually 5%.

s

Sample standard deviation.

s^2

Sample variance.

Distribution of sample statistics is less variable than distribution of individual observations from the same population. T/F?

TRUE. The standard error (which measures the spread of the sampling distribution) is always smaller than the sample standard deviation, S E equals fraction numerator s over denominator square root of n end fraction.

Point estimates based on a sample are sometimes far from a parameter's value. T/F?

TRUE. This may be due to non-random (biased) sampling. In addition, even in cases of true random sampling, some (very few, but some) samples may yield sample means more than 3 standard errors away from the mean.

μ

The average of all observations in the population. The population mean.

Effect Size

The difference between the point estimate and the null value.

P Value

The distribution difference from what was observed and what was expected to be observed. If less than 0.5% in magnitude, safer to reject the null hypothesis. P(observed or more extreme outcome l Ho true)

Central Limit Theorem (CLT)

The distribution of sample statistics is nearly normal, centered at the population mean, and with standard deviation equal to the population standard deviation divided by square root of the sample size.

x̅

The mean, or average. Summation of all values divided by the amount of values, n.

Confidence Level

The percentage of times that a confidence interval will contain μ.

Sampling Distribution

The probability distribution of a sample statistic when a sample is drawn from a population.

β

The probability of committing a type 2 error.

Margin of Error

The range of percentage points in which the sample accurately reflects the population.

n

The sample size.

Sample Error

The standard deviation of the sample means. Usually less variable than an individual sample distribution.

Standard error vs standard deviation

The standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean.

Precision

The width of a confidence interval.

More accurate means a higher confidence level. As the CL increases, the interval width also increases. T/F

True.

Which error is worse to make? (According to William Blackstone)

Type 1 Error. "Better that ten guilty persons escape than the one innocent suffer."

Type 1 Error Rate

We reject the null hypothesis H(o) when the p-value is less than 0.05. For those cases where the null hypothesis H(o) is true, we do not want to incorrectly reject it more than 5% of those times. Increasing α increases the type 1 error rate.

Accuracy

Whether of not the confidence interval contains the true population parameter.

P-Value Formula

Z = (sample statistic - null value) / SE

The significance level and the confidence level are:

complementary; but only in two-sided tests.

Central Limit Theorems for Means Applet

https://gallery.shinyapps.io/CLT_mean/

Normal Probability Table

https://www.openintro.org/download.php?file=os2_prob_tables&referrer=coursera.php

If the p-value is high,

it would be likely to observe the data even if the null hypothesis were true, hence do not reject Ho.

If the p-value is low,

it would be very unlikely to observe the data if the null hypothesis were true, hence reject Ho.

Since compete populations are different (or impossible) to collect data on, we use sample statistics as"

point estimates for the unknown population parameters of interest.

Sample statistics vary from:

sample to sample.

Sampling distributions get closer to normality as:

the sample size increases.

As sample size increases,

variability among the sample means would be lower and the standard error also decreases.

Z score more info

z = (X - μ) / σ where z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation. Here is how to interpret z-scores. - A z-score less than 0 represents an element less than the mean. - A z-score greater than 0 represents an element greater than the mean. - A z-score equal to 0 represents an element equal to the mean. - A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc. - A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc. - If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3. - Here is another way to think about z-scores. A z-score is the normal random variable of a standard normal distribution.