QBA #4

In hypothesis testing, which hypothesis is tentatively assumed to be true?

null hypothesis

Know how to calculate standard error of the mean given n, M, & SD

standard deviation / the square root of sample size = standard error of mean

When to use a z-distribution vs. a t-distribution.

standard deviation of the population IS KNOWN use z-distribution standard deviation of the population is NOT KNOWN use t-distribution

How does increasing degrees of freedom affect the t distribution?

the greater the degrees of freedom, the more closely the t-distribution resembles the standard normal distribution. As the degrees of freedom increases, the area in the tails of the t-distribution decreases while the area near the center increases.

What does 'Po' represent?

the hypothesized value for the population proportion

How does the increasing mean difference affect the likelihood that a hypothesis test will be significant?

the p value decreases, thus making it more likely that we reject the null hypothesis.

observation

value assigned to only one element

What degrees of freedom are used with a t distribution?

when you have a sample and estimate the mean, you have n - 1 degrees of freedom, where n is the sample size. The DF define the shape of the t-distribution that your t-test uses to calculate the p-value.

higher confidence levels (90% to 95%) provide

wider confidence intervals

Which measure of location is used with categorical nominal data?

Mode

variable(s)

A characteristic of interest for the elements. observations that take on different values (columns)

mean

A measure of central location computed by summing the data values and dividing by the number of observations. mathematical center scale

median

A measure of central location provided by the value in the middle when the data are arranged in ascending order. geographic center ordinal

mode

A measure of location, defined as the value that occurs with greatest frequency. most frequently occurring nominal

range

A measure of variability, defined to be the largest value minus the smallest value.

convenience sampling

A nonprobabilistic method of sampling whereby elements are selected on the basis of convenience.

Parameter

A numerical characteristic of a population, such as a population mean 'mu', a population standard deviation 'sigma', a population proportion p, and so on.

Sample statistic

A numerical value used as a summary measure for a sample (e.g., the sample mean,'x-bar' , the sample variance,s-squared , and the sample standard deviation, s). A sample characteristic, such as a sample mean 'x-bar', a sample standard deviation s, a sample proportion 'p-bar', and so on. The value of the sample statistic is used to estimate the value of the corresponding population parameter.

sampling distribution

A probability distribution consisting of all possible values of a sample statistic.

cluster sampling

A probability sampling method in which the population is first divided into clusters and then a simple random sample of the clusters is taken. A probabilistic method of sampling in which the population is first divided into clusters and then one or more clusters are selected for sampling. In single-stage cluster sampling, every element in each selected cluster is sampled; in two-stage cluster sampling, a sample of the elements in each selected cluster is collected.

stratified random sampling

A probability sampling method in which the population is first divided into strata and a simple random sample is then taken from each stratum

systematic sampling

A probability sampling method in which we randomly select one of the first k elements and then select every kth element thereafter. A method of choosing a sample by randomly selecting the first element and then selecting every kth element thereafter.

experiment

A process that generates well-defined outcomes.

random sample

A random sample from an infinite population is a sample selected such that the following conditions are satisfied: (1) Each element selected comes from the same population; (2) each element is selected independently.

sample

A subset of the population.

census

A survey to collect data on the entire population.

What is the role of the critical value in hypothesis testing?

A value that is compared with the test statistic to determine whether 'Ho' should be rejected. establishes the boundary of the rejection region

How does increasing sample size affect the margin of error?

As sample size increases, the margin of error decreases. As sample size decreases, the margin of error increases.

Key difference between binomial and hypergeometric distributions

Both the hypergeometric distribution and the binomial distribution describe the number of times an event occurs in a fixed number of trials. For the binomial distribution, the probability is the same for every trial. For the hypergeometric distribution, each trial changes the probability for each subsequent trial because there is no replacement.

How are the following affected by outliers: mean, SD, range, IQR?

Mean increases with high outlier Mean decreases with low outlier outliers increase the standard deviation. The more extreme the outlier, the more the standard deviation is affected. Range: Spread of data increases Interquartile Range is Not Affected By Outliers

Hypothesized population proportion

Hypothesis tests about a population proportion are based on the difference between the sample proportion 'p-bar' and the hypothesized population proportion 'Po'. The methods used to conduct the hypothesis test are similar to those used for hypothesis tests about a population mean. The only difference is that we use the sample proportion and its standard error to compute the test statistic. The p-value approach or the critical value approach is then used to determine whether the null hypothesis should be rejected.

How does increasing confidence levels affect confidence intervals?

Increasing the confidence level widens the confidence interval. The wider the interval, the more likely that the true parameter will be captured...the margin of error increases

Box plot vs. five-number summary

box plot is the graphical representation of a five-number summary

How does increasing sample variance affect the likelihood that a hypothesis test will be significant?

It will increase the estimated standard error and decrease the likelihood of rejecting 'H0' null hypothesis.

Null hypothesis vs. alternative hypothesis

Null hypothesis The null hypothesis is a general statement that states that there is no relationship between two phenomenons under consideration or that there is no association between two groups. The null hypothesis believes that the results are observed as a result of chance. It is the hypothesis that the researcher tries to disprove. The result of the null hypothesis indicates no changes in opinions or actions. If the null hypothesis is accepted, the results of the study become insignificant. If the p-value is greater than the level of significance, the null hypothesis is accepted. The null hypothesis allows the acceptance of correct existing theories and the consistency of multiple experiments. Alternative hypothesis An alternative hypothesis is a statement that describes that there is a relationship between two selected variables in a study. The alternative hypothesis believes that the results are observed as a result of some real causes. It is a hypothesis that the researcher tries to prove. The result of an alternative hypothesis causes changes in opinions and actions. If an alternative hypothesis is accepted, the results of the study become significant. If the p-value is smaller than the level of significance, an alternative hypothesis is accepted. Alternative hypothesis are important as it establishes a relationship between two variables, resulting in new improved theories.

descriptive statistics

Tabular, graphical, and numerical summaries of data. numbers: central tendency and variability

Elements

The entities on which data are collected.

element

The entity on which data are collected. subjects or participants (row)

sampling error

The error that occurs because a sample, and not the entire population, is used to estimate a population parameter. Difference between sample static and actual population parameter.

Data

The facts and figures collected, analyzed, and summarized for presentation and interpretation.

What is the mean of the t distribution? The standard deviation?

The mean of the distribution is equal to 0; standard deviation is unknown

statistical inference

The process of using data obtained from a sample to make estimates or test hypotheses about the characteristics of a population.

When computing margin of error requirements, what planning value of p should you use?

The range of p is 0 to 1, and therefore the range of p(1-p) is 0 to 1. The value of p that maximizes p(1-p) is p=0.5. Consequently, if there is no information available to approximate p, then p=0.5 can be used to generate the most conservative, or largest, sample size.

population

The set of all elements of interest in a particular study. The collection of all elements of interest. (Cenus)

How does decreasing p-values affect the evidence against 'Ho' null hypothesis?

The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).

How does increasing sample size affect the parameter estimates?

The standard error is dependent on sample size: larger sample sizes produce smaller standard errors, which estimate population parameters with higher precision.

Type I error vs. Type II error

Type I error Type 1 error, in statistical hypothesis testing, is the error caused by rejecting a null hypothesis when it is true. The probability of type I error is equal to the level of significance. It can be reduced by decreasing the level of significance. It is caused by luck or chance. Type I error is associated with rejecting the null hypothesis. It happens when the acceptance levels are set too lenient. Type II error Type II error is the error that occurs when the null hypothesis is accepted when it is not true. The probability of type II error is equal to one minus the power of the test. It can be reduced by increasing the level of significance. It is caused by a smaller sample size or a less powerful test. Type II error is associated with rejecting the alternative hypothesis. It happens when the acceptance levels are set too stringent.

If a hypothesis is rejected at a 5% level of significance, must it also be rejected at 10%? At 1%?

Yes; No

Know how to compute a mean

add up all the numbers, then divide by how many numbers there are

When is a standard deviation equal to zero?

all of the values in the sample are identical, the sample standard deviation will be zero

outliers

extreme vaules central tendency: skews mean toward outlier, minimal effect on median, none on mode variability: increases variance and standard deviation range is most susceptible to outliers IQR not at all

Where will you find the rejection region in a one-tailed hypothesis test?

if you wanted to be 95% confident that your results are significant, you would choose a 5% alpha level (100% - 95%). That 5% level is the rejection region. For a one tailed test, the 5% would be in one tail. For a two tailed test, the rejection region would be in two tails.

Research: how to we apply findings from a sample to a population?

inferential statistics - data from a sample are used to make an inference about a population.

What is the expected value of a random variable?

is a summary statistic, providing a measure of the location or central tendency

Which measure of location is used with continuous numeric data?

mean, median, and mode are equal.