Chapter 8/9 Verbal Exam Questions
Explain the difference between a one-sided alternative hypothesis and a two-sided alternative hypothesis?
A one-sided hypothesis claims that a parameter is either larger or smaller than the value given by the null hypothesis. A two-sided hypothesis claims that a parameter is simply not equal to the value given by the null hypothesis -- the direction does not matter.
What is alternative hypothesis?
A statement that directly contradicts the null hypothesis. Determine whether or not to accept or reject this statement based on the likelihood of the null (opposite) hypothesis being true. Can include a not equal to, greater than or less than sign
Explain the difference between a Type I Error and a Type II Error.
A type I error (false-positive) occurs if we reject a null hypothesis that is actually true in the population; a type II error (false-negative) occurs if we fail to reject a null hypothesis when the alternative is true
What happens to the margin of error as n increases? By how many times must the sample size n increase in order to cut the margin of error in half?
As n increases, the margin of error decreases. The sample size must be 4x as large to cut the margin of error in half
In a sampling distribution of x̅, why is the interval of numbers between x̅±2s called a 95% confidence interval ?
Because 95% of the Normal population is within 2 standard deviations of the mean
Why is it best to have high confidence and a small margin of error?
Because it gives a higher chance your interval captures the true value and that the interval would be narrower. it'll give a closer approximation to what were estimaing for the true mean value.
Why do we need to be careful about multiple analyses?
Because of producing significant results by chance alone
Describe the differences between a standard normal distribution and a t distribution.
The t-distribution has a larger area in the tails and a t distribution is wider in the tails
Describe the similarities between a standard normal distribution and a t distribution.
They have similar shape, they're symmetric and has bell shaped curves
What is a T-distribution? Why do we use it instead of Z when calculation significance tests for a mean?
When you draw an SRS of a size n from a large population that has a normal distribution with mean and standard deviation and use it instead to compare the sample with population mean
What are the p̂, P, and σ used for constructing a confidence interval for a proportion?
p̂ - proportion in the sample P - p-value is the probability which is the result of such a statistical test σ - population standard deviation
Where do p̂, P, and σ come from?
p̂ - the sample p - population σ - population
What is the standard error of the sample mean x̅?
s over the square root of n
How small should the P-Value be in order to claim that the result is statistically significant?
smaller than alpha
What is the standard deviation of the sample mean x̅?
standard deviation over square root of N
Describe how to calculate the power of a significance test.
1-β, probability of making a type II error
Under what assumptions is s a reasonable estimate of σ ?
1.) If it's an SRS 2.) If it meets the 10% condition 3.) If the sampling distribution meets the normal and large counts conditions
What are the three ways you can increase the power of a significance test? Explain how each improves the power of a significance test
1.) Increase the sample size - decreases the spread of both null and alternative distributions which decreases the amount of overlap between the two distributions, making it easier to detect a difference between the null and alternative parameter values 2.) Increase the significance level alpha - increases the area of reject H0 in both the null and alternative distribution 3.) Increase the difference between the null and alternative parameter values that is important to detect
What are the conditions for constructing a confidence interval for a proportion?
1.) Random: The data came from a well-designed random sample or randomized experiment - 10%: When sampling without replacement, check that n ≤ 1/10N 2.) Large Counts: Both np̂ and n(1-p̂) are at least 10
What is the value of z* for a 90% confidence interval? Include a sketch.
1.645
What is the value of z* for a 95% confidence interval? Include a sketch.
1.96
What is the value of z* for a 99% confidence interval? Include a sketch.
2.576
What is meant by a margin of error?
How far "off" you are from the true mean or parameter
If a P-Value is large, what do we conclude about the Null hypothesis?
If it's larger than alpha, we don't have enough evidence to reject the null hypothesis
Why might statistical significance not be the same thing as practical significance?
If the study is based on a very large sample size, relationships found to be statistically significant may not have much practical significance. Almost any null hypothesis can be rejected if the sample size is large enough
What happens to the t distribution as the degrees of freedom increase?
The t distribution and curve becomes closer to a normal distribution and a normal density curve
What does a test statistic estimate?
It measures how far a sample statistic diverged from what we'd expect if the null hypothesis is true.
What is meant by a significance level?
It's a fixed value to compare to the p-value to determine whether an observed result it likely or not. It contributes to the probability of a type 1 error
How do we do a significance test for paired data?
Making comparisons by analyzing differences in each pair and if the conditions for interference are met, use t-procedures to perform inference about the mean difference
Define a level C confidence interval
Means for a parameter to have two parts -An interval calculated from the data, usually of the form: point estimate ± margin of error - A confidence level C, which gives the probability that the interval will capture the true parameter value
What are the conditions required to do a significance test for a proportion?
Random- SRS - 10%: when sampling without replacement, check n≤1/10N Large Counts: both np̂ and n(1-p̂) are at least 10
What factors should be considered in using significance tests wisely?
Sample size, statistical significance and practical importance, multiple analysis
Sketch and label a 95% confidence interval for the standard normal curve
Sketch a normal curve with the middle shaded and a small area in both tails with each having 0.025 not shaded
Sketch and label a 90% confidence interval for the standard normal curve.
Sketch a normal curve with the middle shaded and a small region in both tails, each having an area 0.05 not shaded
Describe the four step process for estimating a population mean
State: Null and alternative hypothesis Plan: Check conditions Do: If the conditions are met, perform calculations Conclude: Interpret your interval in the context of the problem.
Explain the four step process for construction of a confidence interval for a population proportion.
State: What parameter do you want to estimate and at what confidence level ? Plan: Identify the appropriate inference method. Check conditions Do: If the conditions are met, perform calculations Conclude: Interpret your interval in the context of the problem.
What are the conditions required to do a significance test for a mean? Show the four step process.
State: null and alternative hypothesis Plan: Check conditions/graph Do: calculations of p-vaue and t statistic Conclude: data
What does the four step process for the write-up of a significance test for a proportion look like?
State: null and alternative hypothesis, significance level and parameters Plan: Check conditions Do: If the conditions are met, perform calculations: Compute the test statistic and find the p-value Conclude: Make a decision about the hypothesis in the context of the problem
What does it mean to be 90% confident?
That after many trials, 90% of the time we will capture the true value/parameter
How do you interpret an 80% confidence interval?
That you're confident at 80% that the real value is in the interval
What does z* represent?
The critical value of z (the # of standard deviations)
The z-Table gives the area under the standard normal curve to the left of z. What does the t-Table give?
The critical values of t or area outside the confidence interval of a t distribution
What is a null Hypothesis?
The default claim that we assume to be true and one we're seeking evidence against A statement about a population parameter and test the likelihood of this statement being true in order to decide whether to accept or reject our alternative hypothesis. can include =, greater than or equal to, or less than or equal to
The formula used to determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m is z* times standard deviation/square root of n ≤m. Solve for n.
The formula used to determine the sample size n that will yield a confidence interval for a population m
What happens to the margin of error as σ decreases?
The margin of error decreases and confidence decreases
What happens to the margin of error as z* decreases? Does this result in a higher or lower confidence level?
The margin of error gets decreases and confidence descreases
In general what is meant by standard error of a statistic?
The result of when the standard deviation of a statistic is estimated from the data
In general, what is meant by the standard error of a statistic?
The result of when the standard deviation of a statistic is estimated from the data
What is the Relationship Between the significance level a and the probability of Type I Error
The significance level is the probability of a type 1 error occurs
How do you calculate the degrees of freedom for a t distribution?
found by subtracting one from sample size: df = n-1
In statistics, what is meant by the P-value?
the probability the statistic would take a value as extreme or more extreme than the one observed. the probability that our observed measurements can occur given that the null hypothesis is true
In statistics, what is meant by a 95% confidence interval ?
we are 95% confident that the interval from blank to blank captures the true proportion of the statistic we're estimating
If a P-Value is small, what do we conclude about the null hypothesis?
we reject the null hypothesis
How would you construct a level C confidence interval for μ if σ is unknown?
x-bar plus or minus t* (s over square root of n)
What are the x̅, μ, and σ used for constructing a confidence interval for a mean?
x̅ - plus or minus a margin of error μ - when population standard deviation is known, the formula is x-bar plus or minus z* times standard deviation over square root of n where x bar is the sample mean σ - divide by square root of n to get margin of errror
Where do x̅, μ, and σ come from?
x̅ - sample mean μ - population mean σ - population standard deviation
Explain how to find a level C confidence interval for an SRS of size n having unknown mean μ and known standard deviation σ.
x̅ ± z* (σ/square root of n)
