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Stats Exam 3
| Question | Answer |
|---|---|
| A probability distribution for a continuous random variable x is specified by ___? | A curve called a density curve |
| The function that specifies this curve denoted f(x) and is called the | Density function |
| What are 2 properties of Density Curve / Function? | f(x) is equal to or greater than 0 The total area under the density curve is equal to 1 |
| The probability that x falls in some interval is | The area under the density curve and above the interval |
| Uniform Distribution is ____? | Constant over an interval and the density curve is a horizontal line (Desmos: uniformdist(minimum, maximum)) --> find cumulative probability to find area |
| How do you find an Expected Value from a Probability Model | Multiply the outcomes by its probabilities and add them up (Be able to do) |
| Find the Probability of a Triangle's Area (Picture) | Look at Pic |
| Why are Normal Distributions widely used? | They approximate many other distributions and they are important in inferential methods (learned later) |
| The standard normal distribution is the normal distribution with ___? | mean = 0 and standard deviation = 1 |
| The density curve is called the ___? | Standard normal curve (or z-curve) |
| How do you find Cumulative Probability on Desmos? | normaldist(mean,standard deviation), click Find Cumulative Probability and enter minimum and maximum as needed |
| How do you Identify Extreme Values? | To find value a with P (z < a) = area, use Desmos: 2nd and Vars then invNorm( ---> Enter the area, 0 for mean, and 2 for standard deviation (Pic with 0.020000 answer) (If it asks for a percentage of values, use 100 - percentage / Pic) |
| How do you Add Extreme Values? | If it asks for the top and low percentage, you can do inversecdf (0.025) and inversecdf (1 - 0.025) |
| What is a Binomal Experiment? (Step by step process) | There are a fixed number of trials Each trials has only two possible outcomes (success and failure) Outcomes of different trials are independent The probability of success is the same for each trial |
| What is the Bionomial Random Variable? | x = number of sucesses observed |
| The probability distribution of x is called the ____? | Binomial probability distribution |
| Example of a non-bianomial experiment? | Pic of probability of picking balls |
| How do you calculate P (x = a) for Bionomial Distribution? | Desmos: binomialdist(trials,probability), find cumulative probability, min and max (Example with coin flip / 50, .5 pic) |
| How do you calculate P(x less than or equal to a) for Bionomial Distribution? | Desmos: binomialcdf(trials,probability,x value) (Example: binomialcdf(50,.5,26) |
| Binomial Distribution Example? | Plant assembly line example |
| What is sampling with Replacement? | Pick an object from a population, Determine if it a success, Put it back, Repeat (x = number of successes has a binomial distribution) |
| What is sampling without Replacement? | Pick an object from a population, Determine if it a success, Don’t put it back, Repeat (x = number of successes is NOT a bionomial distribution) |
| For a bionomial random variable, you find mean and standard deviation with ____? | Two pics |
| Suppose you take a multiple choice exam with 25 questions and each question has 5 choices..... | 3 Pics |
| Regarding the proportions of a population that has some characteristic, an individual or object that has that characteristic is a ____ and is a ____ otherwise | Success & Failure |
| The sample proportion of successes, p̂, is ____? | p̂ = number of successes in the sample / n (size of a random sample for the population) |
| Population Proportion is denoted by ____? | p |
| How do you find Variance? | n x p x (1 - p) |
| Population Proportion Example | See pics |
| What is Sampling Variability? | The observed value of a statistic varies from sample to sample depending on the particular sample selected |
| What is Sampling Distribution? | The distribution formed by the values of a sample statistic for every possible different sample of a given size |
| What are the General Properties of Sampling Distribution? | 1. mean = p (mean of the sampling distribution for p̂ is p) 2. standard deviation = square root of p(1- p) / n 3. When n is large and p is not too near 0 or 1, the sampling distribution of p̂ is approximately normal |
| As n increases, the standard deviation ____? | Decreases |
| What is Standard Error? | The standard deviation of a sampling distribution |
| What is an Example of Solving for the mean and standard deviation of a sampling distribution of p̂? | Look a pic involving 20% of vehicles and 45 vehicles |
| The farther the value of p from .5, the ____? | larger n should be for p̂ to have a sampling distribution that is approximately normal (np greater than or equal to 10 and n(1 - p) is greater than or equal to 10 is normal sampling distribution) |
| is n = 20 or n = 50 with a proportion of 56% considered an approximately normal distribution? | n = 50 ----> 50(.56) = greater than 10 and 50(1-.56) = greater than 10 |
| Know the notations for mean and standard error | Look on canvas |
| Suppose p = 0.72 and we take a sample with n = 100 | 4 Pics |
| Know notation for population mean, population standard deviation, sample mean, and sample standard deviation | canvas |
| x̄, or sample mean, denotes ____? | the mean of the observation in a random sample of size n from a population with mean u and standard deviation σ |
| Rules of Sampling Distribution | 1. mean of sampling distribution = mean of the population 2. standard deviation of a sample = standard deviation / square root of n 3. When the population distribution is normal, the sampling distribution of x̄ is also normal for any sample size n |
| 4th Rule of Sampling Distribution | (Central limit theorem) When n is large, the sampling distribution of x̄ is approximated by a normal curve, even when the population distribution is not normal (can safely be applied if n is greater than or equal to 30) |
| How to find Standard Error for 2 Sampling Distributions | square root of p(1- p) / n + p(1- p) / n (Both are for each population and are square rooted after both are added) |
| What is QSTN? | Question Type (estimation), Study Type (sample data), Type of data (categorical), and Number of Samples (one sample) |
| What is Margin of Error? | The maximum likely estimation error --> It is unlikely that an estimate will differ from the actual value of the population characteristic by more than the margin of error |
| What is Confidence Level? | With a confidence interval is the success rate of the method used to construct the interval |
| If n is large, 95% of all possible random samples will produce a value of p̂ that is within _____? | 1.96 square root of p (1 - p) / n |
| Margin of Error Conditions | 1. The sample is a random sample 2. The sample size is large enough - np̂ is greater than or equal to 10 AND n (1 - p̂) is greater than or equal to 10, OR sample has at least 10 successes + 10 failures |
| How to Solve for Margin of Error | 1.96 square root of p̂ (1 - p̂) / n |
| What is a Confidence Interval? | An interval that you think includes the value of the population characteristic |
| What is a Confidence Level? | The confidence level associated with a confidence interval is the success rate of the method used to construct the interval |
| If we construct a 95% confidence interval, _____? | Different random samples will give you a different CI and 95% of the intervals will capture the true value of the population characteristic |
| For 95% of samples, p will be in the interval ____? (How to solve for Confidence Interval / Length) | (p̂ - margin of error, p̂ + margin of error) or p̂ + or - margin of error |
| What is the Method for Estimation (EMC3)? | Estimate (Find p) Method (QSTN) Check (Random? Is n large?) Calculate (p̂ + or - 1.96 square root of p̂ (1 - p̂) / n) Communicate results |
| How do you solve for CI 95%? | p̂ + or - 1.96 square root of p̂ (1 - p̂) / n |
| 90% of values are ______ standard deviations from the mean | 1.645 |
| 99% of values are ______ standard deviations from the mean | 2.58 |
| To construct a 90%, 95%, or 99% confidence interval, use the following (assuming all other conditions are met): | p̂ + or - (z-critical value) square root of p̂ (1 - p̂) / n |
| Confidence Interval Width Rules | 1. The greater the confidence level, the wider the interval 2. The greater the sample size, the narrower the interval 3. The closer p̂ is to .5, the wider the interval |
| Suppose you want your margin of error to be .01 . . . what is the equation to solve? | 1.96 square root of p̂ (1 - p̂) / n = .01 OR, using a 95% confidence level, you can always solve as n = p(1 - p) (1.96 / M) squared *M = margin of error and p = population proportion |
| If p is unknown, use ____? | p = 0.5 |
| What is QSTN for two indepedent samples? | Question type: Estimation Study type: Sample data Type of data: One categorical variables Number of samples: Two independent samples |
| How to Solve for Confidence Interval with Two Independent Samples | statistic (p̂1 - p̂2) plus or minus (critical value) (standard error) |
| How to solve for Standard Error? | square root of p̂1 (1 - p̂1) / n1 + p̂2 (1 - p̂2) / n2 |
| If both endpoints are positive _____? | likely p1 > p2 (interval gives by how much) |
| If both endpoints are negative _____? | likely p1 < p2 (interval gives by how much) |
| If 0 is included in the interval _____? | plausible that p1 = p2 |
| the t distribution corresponding to any particular degree of freedom (df) is _____? | bell shaped, centered at 0 |
| Every t distribution has greater variability _____? | than the standard normal distribution |
| As the number of degrees of freedom increases _____? | the variability of the corresponding t distribution decreases (df = n -1) |
| As the number of degrees of freedom increases, the corresponding sequence of t distributions _____? | approaches the standard normal distribution |
| How to find T Critical Numbers (TI) on Desmos * df = n - 1 | tdist(sample size - 1).inversecdf(1- confidence interval decimal percent) / 2 (only divide the (1 - confidence interval, which can be like .95)*t critical number will be the opposite sign of the result (Pic) |
| Solving for T Critical Numbers practice | 2 Pics, 3 answers |
| Conditions for Confidence Interval for 1 Mean | The sample is a random sample from the population of interest or is otherwise representative The sample size is large (n is equal to or greater than 30) OR the population distribution is approximately normal |
| How to Solve for Confidence Interval for 1 Mean (Also can use Canvas tool for Desmos) | x̄ + or - (t critical value) (standard deviation divided by the square root of n) *t critical value based on df = n - 1 and the confidence interval |
| The sample size required to estimate a population mean at the 95% confidence level is u with a specified margin of error M is _____? | n = (1.96 x standard deviation then divided by M) squared |
| If standard deviation is not known, it can be estimated from _____? | previous information or range/4 (as long as the data is not too skewed) |
| A hypothesis is ____? | a claim or statement about a one (or more) population characteristic (Examples: p = .50, u < 102, p1 - p2 > 0) *hypotheses are always about the population characteristic (not a statistic) |
| A hypothesis test uses _____? | sample data to choose between two competing hypotheses about a population characteristic |
| A Null Hypothesis, denoted, H0, is ____? | a claim about a population characteristic that is initially assumed to be true |
| The Alternative Hypothesis, denoted Ha, is ____? | a competing claim |
| To reject H0, there is ____? | convincing evidence against the null hypothesis If H0 were true, the data would be very surprising |
| A hypothesis test can only demonstrate strong support for the _____? | alternative hypothesis (So we will translate our claim into an alternative hypothesis) |
| How to solve for hypothesis test? | List the population characteristic tests Pics (4) |
Created by:
AnaHuls