Download Stat 226 SI – Review Worksheet – Answer Key 4/8/13 Mark each

Stat 226 SI – Review Worksheet – Answer Key
4/8/13
Mark each statement as true or false. If the statement is false, explain why.
____ Voluntary response surveys tend to have a large amount of bias because those that typically
respond feel strongly about the topic of the survey. True
____Sampling the height of every fifth person in line for a rollercoaster at Adventureland is an example
of stratified random sampling. False -- This example is characteristic of systematic sampling. Sampling
every 5th, 10th, 20th, etc. person is a system by which a sample is taken.
____The margin of error is equal to the width of a constructed confidence interval. False -- The margin
of error is equal to half the width of a confidence interval. The formula for a confidence interval is 𝑥̅ ±
𝑧∙
𝜎
..
√𝑛
The margin of error is equal to 𝑧 ∙
𝜎
..
√𝑛
The same is true for the formula that uses a t-critical
value.
____ To construct a confidence interval, the sampling distribution of the sample mean has to follow at
least approximately a normal distribution and the sample taken has to be taken randomly. True – these
are the 2 necessary assumptions for constructing confidence intervals.
____ The Law of Large Numbers states that, when a population distribution is symmetric and bellshaped (but not normal), the sampling distribution of the sample mean will approach normality for
moderately large sample sizes. False – the Law of Large numbers states that, as we increase the sample
size, n, 𝑋̅ approaches the true sample mean µ. The above statement is a portion of the Central Limit
Theorem.
____ The alternative hypothesis is the generally accepted belief. False – The null hypothesis is the
generally accepted belief.
____ As a sample size increases, a confidence interval gets wider, and we can, therefore, predict our
population mean more precisely. False – As the sample size increases a confidence interval narrows
because the margin of error decreases. Therefore, we can predict our population mean more precisely.
_____________________________________________________________________________________
Define the Central Limit Theorem: When a population distribution is symmetric and bell-shaped (but
not normal), the sampling distribution of the sample mean will approach normality for moderately large
sample sizes (n>15). When a population distribution is neither symmetric nor bell-shaped (it is not
normal), the sampling distribution of the sample mean will be approximately normal for large sample
sizes (n > 30).
What is the difference between a sample statistic and a population parameter? (It may help to give
examples.) A statistic is a numerical summary, such as 𝑥̅ , that describes a sample. A parameter is a
numerical summary, such as µ, that describes a population.
You are studying a marketing strategy used by restaurants in one of your classes this semester. This
marketing strategy involves reporting prices with and without a dollar sign. Researchers believe that
restaurants see an increase in sales for menus that do not use the dollar sign. You want to survey
business students at Iowa State University to see how often (what percentage of the time, or the rate
of success) they think this strategy would work. After randomly collecting responses from 100
different students in a simple random sample, you obtain a mean of 75% and standard deviation of
5%. Do the following:
Describe the sampling distribution.
We are not given any information about the population distribution, concerning its normality.
Regardless, we know that the sample size is sufficiently large (n = 100), so we can assume the population
distribution is approximately normal. The center is at , 𝑥̅ = 75, and the spread is equal to 𝜎 = 5.
Construct a 95% confidence interval for the true mean rate of success business students at Iowa State
University believe this strategy actually attains.
𝑥̅ ± 𝑡 ∗
𝑠
√𝑛
Df = 100-1 = 99…… t = 1.990 (round down to be conservative)
75 ± 1.99 ∗
5
100
√
(74.005, 75.995)
We can be 95% confident that the true mean success rate is between 74.005% and 75.995%.
If we know the population mean rate of success to be equal to 72% and the standard deviation of the
population to be 4%, what is the probability of obtaining a sample mean greater than 75.5%?
𝑃(𝑋̅ > 75.5)
𝑧=
𝑥̅ −𝜇
𝜎
=
75.5−72
4
= 0.875 which is approx. = 0.88
P = 1- 0.8106 = 18.94%
The Cyride Orange bus is scheduled to make a full cycle from the Commuter Lot up to campus and
along its route and back in 17 minutes. In order to monitor the efficiency of the Cyride schedule and
bus drivers, the Cyride managers collect a sample of 40 cycles every week. They use this data to test if
the Orange bus actually makes a full cycle along its route in 17 minutes or not. Propose the
appropriate null and alternative hypothesis they need in order to do this test. Where do we place the
benefit of the doubt?
H0 : µ = 17
HA : µ ≠17