Statistics Exam 2

The IQ level of students at a particular university has an unknown mean. A simple random sample of 100 students is found to have a sample mean IQ of x̄ = 115 and a sample standard deviation of s = 15. Calculate a 95% confidence interval for the mean IQ level of all students in this university. Suppose a sample of size 250 was taken instead of size 100. How will the margin of error change? If the researcher wanted to have 95% confidence in the results with a margin of error of 5.1, how many students must be sampled?

(112, 118) the margin of error will decrease in size 35

Four steps for tests of significance

--state (specify claims about parameters of interest) -- plan (choose procedure, specify Hnot, Hsuba, alpha) -- solve (check conditions, calculate test statistic and p-value) -- conclude (compare p-value to alpha, interpret test results)

Estimator

-A general statistic that estimates the parameter. -Exp. estimator of mu is xbar

Cons of point estimation

-Always wrong, because it measures sample, not population

Statistical dogma of process

-all processes have natural variation (raw material, human performance, equipment performance, measurement) -all processes susceptible to unnatural variation (broken machine, bad batch of raw material, poorly trained operator)

What does sampling distribution describe?

-all sample means from all possible random samples of the same size taken from the same population

Why care about sample distribution?

-allows us to assess uncertainty of sample results -if we knew spread of the sample, we would know how far our xbar may be from reality -about 95% of the time, the sample mean will be within 2 standard deviations of the population mean.

key ingredients of statistical inference

-conclusion about parameter -measure of uncertainty

Xbar as an estimator for mu

-good if sampling is done randomly and is unbiased -sample size increases, the accuracy of xbar increases (smaller standard deviation)

Central Limit Theorem

-if you take a large SRS of size n from a sample the shape of the sampling distribution will approximately be normal. -shape gets more normal as n increases -n>= 30 is considered large -allows use of the standard normal table to approximate probabilities of xbar

Margin of error

-likely maximum difference between statistic and parameter at stated confidence level -accounts for uncertainty due to sampling variability

Test of signficance

-objectively answers questions: is the observed difference from the claim, real, or chance?

Out of control signals

-one point above or below the control limits. -run of nine points above or below the mean (as unlikely as one above or below line) -as soon as this is observed. STOP the process

Process

-series of interconnected steps in producing a product or service

four steps for confidence intervals

-state (specify parameter of interest)--plan (choose procedure, level of confidence)--solve (check conditions, carry out procedure)--conclude (interpret confidence interval)

Problem solving steps for t confidence interval

-state parameter you are interested in -plan procedure (t confidence interval) with confidence interval -plan sample size -solve. Check that you can do confidence interval 1. SRS and 2. normal or large sample size. Calculate confidence interval using Xbar +- tstar x s/square root n -conclude in the context of the problem

properties of t distributions (distributions from s)

-symmetric -bellshaped -mean = 0 -the smaller the df, the larger the spread (more uncertainty) -the larger the df, the closer the t-distribution to the standard normal.

How to calculate confidence interval if sigma is unknown?

-use s instead of sigma -use t star instead of zstar.

statistical process control

-use statistical paradigm to monitor process variables over time to determine if variability is consistent with normal variation. -if consistent, continue process -if inconsistent, stop process and find cause of unnatural variation

when to use confidence intervals

-when we collect data randomly -when sample size is large or population distribution is normal

four elements of tests of significance

1. Claim 1 and claim 2: opposing claims about an unknown parameter. Presumption is for claim 1 unless there is strong evidence against it. 2. Outcome: standardized outcome that measures how far the outcome diverges from claim 1 3.Assessment of evidence: howe likely is it to get this outcome if claim 1 is true 4.conclusion: an outcome that would rarely happen if claim 1 is true is good that claim 1 is not true; hence we believe claim 2 to be true

statistical inference

1. Draw conclusion about parameter using a statistic 2. With a measure of uncertainty

Two forms of statistical inference

1. confidence interval to estimate a parameter. 2. Test of significance to assess claim about a parameter

Based on sample results, a 90% confidence interval for the mean servings of fruit per day consumed by grade school children is (0.21, 2.45). What is the margin of error?

1.12

What is the z* associated with 90% confidence?

1.645

A biologist wishes to estimate the mean number of teeth in an adult tiger shark. He wishes to generate a 99% confidence interval with a margin of error of 4 teeth. The standard deviation, σ, is known to be 21.3. How many tiger sharks must he sample?

189

degrees of freedom for sample size 20

19 (n -1)

What is the t* associated with 98% confidence and df = 37?

2.457

Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the mean of the sampling distribution of x-bar?

80

The sampling distribution of x-bar gives _______ from all possible samples of the same size from the same population.

All x-bar values

If we take random samples of size 75 from this population, what will the shape of the sampling distribution of x̄ be?

Approximately Normal

Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the shape of the sampling distribution of x-bar?

Approximately Normal

According to the Central Limit Theorem, for random samples, what is the approximate shape of the sampling distribution of x-bar when the population distribution is non-Normal?

Approximately Normal if the sample size is large

An insurance agent collects a random sample of n = 47 insurance premiums and finds their average and standard deviation. What is the population and statistic in this study?

Average amount of all auto insurance premiums; average amount of n = 47 auto insurance premiums

Interval estimation

Based on a sample of n = 47 policies, we estimate that the average claim of all accidents is between $1700 and $1900

shape

Case 1: population normal will have a distribution of xbar that is normal Case 2: population non-normal will have a normal distribution if n is greater than or equal to 30 (central limit theorem)

What is the quantity z*?

Confidence multiplier

Increasing the sample size will lead to a wider margin of error. T or F

False

T or F: The purpose of a confidence interval is to estimate the value of a sample statistic.

False

T or F: When the population standard deviation, σ, is unknown, we cannot compute a confidence interval.

False

One of your professors claims 90% of BYU students are currently enrolled in a religion course. To test this claim, you randomly sample 300 BYU students and find that only 78% of them are enrolled in a religion course. Based on these sample results, you have evidence against your professor's claim. What type of statistical inference did you use?

Hypothesis testing

All students in the US who took the ACT in 2014 had a mean score of μ = 21.0. Suppose you randomly select two samples of students from this population, and you calculate the sample mean for each. Sample 1 has a size of n = 40, and Sample 2 has a size of n = 250. Which sample is more likely to get a sample mean of 18 or less?

Sample 1 is more likely.

What is the advantage of reporting the average of several measurements rather than the result of a single measurement?

The average of several measurements is more likely to be close to the true mean than the result of a single measurement.

Hypothesis testing

The insurance agent believes that the average accident claim amount at her agency is $2500. Based on the sample n = 47 claims, we found that the average claim amount was $1800. This data, therefore provides evidence that the average claim amount is less than $2500. That is, xbar = $1800 is an outcome that would rarely happen if the average was indeed $2500

properties of confidence interval

The margin of error (m) controls the width of the interval -as sample size increases, m and width decreases -as confidence increases, m and width increases

Suppose the correct answer is 56 sharks (it isn't), but the researcher can only afford to sample 25 sharks. If he wishes to maintain a 99% confidence level, what effect will this have on the resulting confidence interval?

The margin of error will be larger, resulting in a wider interval.

A student takes a random sample of freshman at BYU and records their age at their first kiss. He calculates a 95% confidence interval of (15.5, 21.2). What parameter is the student trying to estimate? Can we say that 95% of the ages are included in the interval (15.5, 21.2)?

The mean age at first kiss for all BYU freshman. No

If we take samples of size 200 rather than 75, what will happen to the mean of the sampling distribution of x̄?

The mean will stay exactly the same

If all possible samples of size 80 are taken from a population, instead of samples of size 20, how would this change the mean and standard deviation of the sampling distribution of x̄?

The mean would stay the same and the standard deviation would decrease.

what does s/square root of n estimate?

The standard deviation of the sampling distribution of x-bar

If we take samples of size 200 rather than 75, what will happen to the standard deviation of the sampling distribution of x̄?

The standard deviation will decrease

Increasing the confidence level will lead to a wider margin of error. T or F

True

Suppose we take all possible samples of the same size from a population and for each sample, we compute x-bar. The mean of these x-bar values will be exactly equal to the mean of the population (μ) from which the samples were taken.

True

Suppose we take all possible samples of the same size from a population and for each, we compute x-bar. The standard deviation of these x-bar values will be less than or equal to the standard deviation of the population from which the samples were taken.

True

How to find degrees of freedom when a number is not on the t table?

always round down, even if number is 999.

Point estimation

based on a sample n = 47 policies, we estimate the average claim amount of all accidents is approximately $1800

proper interpretation of a confidence interval requires three things:

confidence level parameter in context calculated interval

Where does 1.96 come from?

corresponds to middle 95% of normal distribution using the table of standard normal probabilities

t or f: The point estimate will always equal the parameter

false

T or F: we can never calculate probalities when a population is skewed

false. CLT

how to use t table

find degrees of freedom.

Center

mean of sampling distribution of xbar or mu

If the sample size changes from n = 144 to n = 200 the point estimator will be ___ accurate

more

spread

standard deviation of sampling distribution xbar = sigma/(square root of n)

What is the name of the quantity s/square root of n?

standard error of x-bar

Sample

subgroup of the population

Which one of the following is the correct representation of the margin of error when sigma is unknown?

t x s/square root of n

Sampling distribution of Xbar

taking the Xbar of 4 values over and over again and placing it on a graph.

proof of contradiction

tests of significance always assumes that claim is false. So if good evidence against claim, then the claim must not be true

Population

the entire group of individuals that are target of our interest

In confidence interval estimation, we use a confidence interval to estimate

the value of a population parameter.

Xbar control chart

tool for monitoring variables of a process, alerting us when unnatural variation seems to have occurred

The symbol for the sample standard deviation is s.

true

Suppose the officers would like a 99% confidence interval rather than a 95% interval using the same data. The 99% confidence interval will be ________ the 95% confidence interval.

wider than

What do we use to estimate μ?

x-bar

What is the symbol for the sample mean?

x-bar

Inference

Drawing conclusions about a population (parameter) based on data from a sample (statistic) with a measure of uncertainty

Consider the following interpretation of 90% confidence level: "90% of all possible confidence intervals computed using the same procedure used to obtain (0.98, 1.02) will contain the value of μ." Is this interpretation of 90% confidence level correct or incorrect? Why or why not?

Correct. It is stated in terms of the confidence interval procedure—not one specific, calculated interval.

Consider the following interpretation of 90% confidence level: "The probability that the mean of all axle diameters lies somewhere in the interval (0.98 cm, 1.02 cm) is 0.90." Is this interpretation of 90% confidence level correct or incorrect? Why or why not?

Incorrect. It states "probability on one specific calculated interval" rather than "confidence in the procedure."

Based on sample results, we are 90% confident that the mean travel time to work for workers 16 and older is between 16.8 and 25.4 minutes. What type of inference is this?

Interval estimation

Suppose we have a very right skewed population distribution where μ = 80 and σ = 20. For random samples of size n = 100, what is the standard deviation of the sampling distribution of x-bar?

Less than 20

When estimating population mean, we can not use "probability", but instead use "confident". Why?

Mu is a fixed number

Consider the following interpretation of 99% confidence: "There is a 99% probability that our interval captures the population parameter." Is this a correct interpretation?

No

Suppose we have a very right-skewed population distribution where μ = 80 and σ = 20. Suppose we take a sample of size n = 10 from this same population. Can we compute the probability that x̄ is greater than 75?

No, because we cannot apply the Central Limit Theorem. Thus, the sampling distribution of x̄ is not normally distributed.

A researcher wishes to estimate the mean amount of money single, undergraduate college students spend on food in a typical month. To generate a sample, she calls the first 120 students listed in the directory of the local college. Is it safe to compute a confidence interval from this sample?

No- sample was not collected randomly

For small random samples from a Normal population distribution, the shape of the sampling distribution of x-bar is

Normal.

When we decrease our sample size and maintain our level of confidence, our margin of error becomes

Wider

Consider the following interpretation of 99% confidence: "99% of intervals calculated with this method will capture the population parameter." Is this a correct interpretation?

Yes

Estimate

a specific value of an estimator. Exp: the average value of the n = 47 claims is $1800

A significant test where Ha is not equal to value in Ho is called

a two-sided test

Sampling distribution results in graphs that have _____ distribution

normal

simple way to find Zstar of confidence intervals

on table sheet

Researchers want to estimate the amount of time teenagers spend watching television during one week. A random sample of 500 teenagers yielded a sample mean of 12.60 hours of television per week. What type of statistical inference is being used?

point estimation

parameter

population values

out of control process

process exhibits unnatural variation over time

in control process

process whose output exhibits only natural variation over time

Consider this formula for a confidence interval when σ is unknown: Which part of this formula is the standard error of x-bar?

s/square root of n

statistic

sample values

Calculating confidence interval

xbar +/- 2(or z) x sigma)/square root of n more specifically, 2 is actually 1.96 standard deviations

What is the symbol for the population standard deviation?

σ