Business Analytics

what affects a confidence interval for the mean’s margin of error?

• level of confidence

• sample size

• variation in the data

What are the characteristics of a simple random sample?

Two Conditions Must Be Met:

• every individual has an equal chance of being selected

• every group of a size (n) has an equal chance of being selected

How could you get an SRS od a size 100?

Give each person a #001-999 and 000 and then randomly choose 100 numbers

What is a parameter?

a characteristic of the population ( population mean ) - often unknown to us

What is a statistic?

an observed characteristic of a sample (sample mean) - calculated

How are statistics and parameters related?

using a statistic to approximate a parameter

Sampling Variation

ex. we want to know how much time CBA Students spend on HW each week : random sample of 30 students - report the sample mean & SD - repeat : do we get the same sample mean and SD? NO there’s variation from sample to sample

Sampling Distribution

ex. sampling distribution of the mean - shows the possible values of a statistic from a population

What does the Central Limit Theorem tell us?

if n is sufficiently large the distribution of sample means is normally distributed

• mean ( same as underlying population )

• SD - SD/*sqrt*N ( called standard error )

We can use CI calculations if :

• the sample is a SRS from the relevant population

• sample size condition (for proportion) holds both np and n(1-p) are larger than 10

Population —> Parameter —> Mean

standard error SD/*sqrt*N

Sample —> Statistic —> Sample Mean

standard error S/*sqrt* N

Student T’s Distribution

• a parameter called degrees of freedom (n-1) controls the shape of the distribution

• as n increases, t-distribution more closely resembles the standard normal distribution

Restrictions

like the confidence interval for the poulation proportion we need to ensure two things :

• representative sample : the sample is a SRS from the relevant population

• Sample size condition : needs to be sufficiently large

z = 1.96

95% confidence level

A precise confidence interval has a

small margin of error

margin of error is affected by

level of confidence

variation in the data

number of observations

What function would you use in excel to calculate z based on CL?

ex. 95% CL so .025 = excel function = NORM.S.INV(.025)

two- sided test

the null hypothesis asserts a specific value for the population parameter (ex. Ho: p = .2 vs Ha: p =/ .2)

one sided test:

the null hypothesis allows any value of a parameter larger (or smaller) than a specific value (ex. Ho: p >= .2 vs Ha: p < .2)

The higher the CL

the higher the margin of error

A manufacturing firm wants to advertise that the majority of customers ( more than 50%) prefer its product over its competitors. The firm collects data from a random sample of consumers. What type of test is this?

Hypothesis test for the proportion (z-test)

percentage —> proportion Ha: p > .50

A random sample of entry-level accountants' salaries nationwide produced a 95% confidence interval ($55,001, $66,501) for the mean salary of an entry-level accountant. Assume all sampling conditions were satisfied. Which of the following is the correct interpretation of this interval?

If the sampling process and interval estimation computation were repeated many times, we would expect 95% of the intervals to contain the actual mean salary for the population of all entry-level accountants in the U.S.

If the 95% confidence interval for the average purchase of customers at a department store is $50 to $110, then $49 is a plausible value for the mean at this level of confidence.

False

All else equal, a 90% confidence interval will be wider than a 95% confidence interval.

False

A proposal to initiate raising funds for a new school building was presented at a town hall meeting. 307 of the 345 townspeople in attendance at the meeting were in favor to the proposal. Assume those in attendance make up a random sample of town members.

A 95% confidence interval for the population proportion of townspeople who are in favor of the proposal is (0.86, 0.92). 

If there had been 1035 people at the meeting and 921 of 1035 had been in favor of the proposal, what would happen to the 95% confidence interval?

The interval width will decrease because the sample size is larger

A random sample of 100 adults is taken from a population of over one million. In order to compute a confidence interval for a mean, if the researchers wanted to decrease the width of the confidence interval, they could: 

Increase the size of the sample

Which of the following affect a confidence interval for the mean's margin of error?

level of confidence

sample size

variation in the data

A pharmaceutical company wants to estimate the percentage of patients for which a particular treatment is successful.

Which method would be most appropriate for this situation?

Confidence interval for the proportion -Since the company just wants an estimate and there are no null or hypothesized values provided, a confidence interval is the most appropriate option.

The alternative hypothesis is the accepted truth or status quo and always includes an inequality (≠,<,or>)

False

A manager is evaluating software to filter SPAM e-mails (cost $15,000). To make it profitable, the software must reduce SPAM to less than 20%. 

Suppose a review of 100 e-mails showed 12% spam. The manager uses this sample of emails to test the following hypotheses: H0: p ≥ 0.2  vs.   Ha: p < 0.2

• The resulting test statistic is z = -2

• The resulting p-value is 0.02275

Which is the correct conclusion?

reject the null hypothesis (H0)

A random sample of 100 adults is taken from a population of over one million. If the researchers
wanted to decrease the width of the confidence interval, they could:
A. Decrease the size of the sample
B. Decrease the size of the population
C. Increase the size of the population
D. Increase the confidence level
E. None of these would decrease the width

E

A researcher wants to know if the majority of women between ages 18 and 25 use a particular dating
app. A sample of 300 women between ages 18 and 25 are asked whether or not they use the this
dating app. What is the parameter of interest and its appropriate statistical notation?
A. One Population Proportion, p
B. One Population Mean, 𝑥̅
C. One Population Proportion, 𝑝̂
D. One Population Mean, μ
E. One Population Standard Deviation, σ

A

A researcher wants to know a person’s heart rate on average after running a 5K. For a sample of 30
runners, the researcher measures each runner’s heart rate immediately after running a 5K race. What
is the parameter of interest and its appropriate statistical notation?
A. One Population Proportion, p
B. One Population Mean, 𝑥̅
C. One Population Proportion, 𝑝̂
D. One Population Mean, μ
E. One Population Standard Deviation, σ

D

Consider a random sample from a population. For which of the following conditions would the
margin of error (and confidence interval) be the smallest?
A. n = 4000 and 90% confidence level
B. n = 8000 and 99% confidence level
C. n = 8000 and 90% confidence level
D. n = 4000 and 99% confidence level

C

Consider a random sample from a population. For which of the following conditions would the
margin of error (and confidence interval) be the largest?
A. n = 1000 and 90% confidence level
B. n = 2000 and 99% confidence level
C. n = 2000 and 90% confidence level
D. n = 1000 and 99% confidence level

D

hich of the following is the correct generic formula for any confidence interval?
A. sample estimate ± standard error
B. sample estimate ± multiplier
C. sample estimate ± multiplier × standard error
D. margin of error ± multiplier

C

In general, which is more likely to contain the unknown population mean?
A. 90% confidence interval
B. 95% confidence interval
C. 99% confidence interval
D. They are all equally likely

C

In general, which is more likely to contain the known population mean?
A. 90% confidence interval
B. 95% confidence interval
C. 99% confidence interval
D. They are all equally likely

A

Which of the following is NOT true about a population parameter?
A. It is a number that describes the population.
B. It can only assume one value since the population is defined.
C. It has a value that may or may not be known once the population is defined.
D. It is a number that describes the sample.

D

Type of Data? Quantitative

population mean —> sample mean —> studetn t’s distribution w/df —> x +- t* S/*sqrt*N

Type of Data? Categorical

population proportion —> sample proportion —> standard normal (z) —> p-har +- z x sqrt p-phat (1-p-hat) / N