STA 101 Chapter Notes - Chapter Unit 3: Statistical Parameter, Alternative Hypothesis, Type I And Type Ii Errors
10 May 2018
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Unit 3 Foundations for Inference
Introduction
Ex: A study shows that the 41% of the public believes that young adults, rather than middle-aged
or older adults, are having the toughest time in today’s economy.
49% of young adults at 18-34 have taken on a job for the sake of paying bills
Margin of error of plus or minus 2.9% for results based on the total sample and 4.4% for adults
ages 18-24 at the 95% confidence level
o We are 95% confident that 38.1% to 43.9% of the public believe that young adults, rather
than middle-aged or older adults, are having the toughest time in today’s economy
o We are 95% confident that 44.6% to 53.4% of 18- year olds have taken a job they didn’t
want just to pay the bills
Addresses sampling variability since we cannot conduct a study on the entire population
Part 1: (1) Sampling Variability and CLT
Suppose we are interested in women’s heights in the US
We cannot calculate the entire population’s height means, so we take sample , women in each
state and take the mean
o The sample of the entire population’s average heights will be more variable than the
sample of the means across the 50 states
o Entire population = sample distribution
o Means of the 50 states = sampling distribution
Sampling distribution mean ≈ population mean
Sampling distribution SD < population SD
o Sampling distribution SD = Standard error
o As the sample size, n, increases, the standard error decreases
▪ Clarification – the size of each sample in the distribution, not the number of samples
in the sampling distribution
Central Limit Theorem: The distribution of sample statistics is nearly normal at the population
mean, and with a standard deviation equal to the population standard deviation divided by the
square root of the sample size.
)f we don’t have access to sigma, which the population standard deviation, then we can use S,
which is the sample standard deviation
Conditions for the CLT
o Independence: Sampled observations must be independent
▪ Can be verified by checking:
• Random sample/assignment
• If sampling without replacement, n < 10% of population
o Sample size/skew: Either population distribution is normal, or if the population distribution
is skewed or if we don’t know what the distribution looks like, the sample size is large
(rule of thumb: n > 30)
▪ The more skewed the distribution, the larger the sample size needs to be
Focus on: If sampling without replacement, n < 10% of population
o Ex: Say you live in a small town with your extended family and )’m sampling for a genetic
characteristic. Say that I take a sample of 10 – it’s very unlikely that your parents and your
relatives are all going to be in the sample. But if I were to take a sample of 500, then it
would be very likely that your parents and relatives are in the sample. We take a small
sample to maintain that the samples are independent of one another.
Focus on: Sample size/skew condition
o The larger the sample, the more the sample resembles a nearly normal distribution
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Part 1: (2) CLT (for the mean) examples
We cannot find a z-score and find the percentile since the distribution is not normal
Distribution is right skewed
o X = length of one song
o P (X > 5) about 17% (eyeballing the bars)
Ex: )’m making a trip that will last a -hour drive. )’m making a playlist of songs. What is the
probability that my playlist will last the entire drive?
o 6 hours = 360 minutes
o P (x1 + x2 + x3 +… + x100 > 360 minutes) = ?
o x̄ = average length of 100 songs
o P (x̄ > 3.6 minutes) = ?
o We will use the CLT, which says that x̄ will be nearly normally distributed with N (mean =
3.45, SE = 1.63/rad(100) = 0.163) N(3.45, 0.163)
o P (x > 3.6) = ?
o Z Score = (3.6 – 3.45)/0.163 = 0.92
o P (Z > 0.92) = 0.179
o There is a 17.9% that my playlist will at least last the entire drive.
Sampling sizes affect shape and spread of distributions
Part 2: (1) Confidence Interval (for a mean)
Confidence interval: a plausible range of values for the population parameter
o )f we report a point estimate, we probability won’t hit the exact population parameter
o If we report a range of plausible values, we have a good shot of finding the actual values
95% confidence interval – Sample mean (x̄ ) 2 SE
o 2 (SE) = margin of error
Ex: A study of 124 couples found that 64.5% turned their heads while kissing. The standard error
associated with this study is 4%. Which of the following false?
o (A) A higher sample size would yield a lower standard error.
o (B) The margin of error for a 95% CI for the % of kissers who turn their heads to the right
is roughly 8%.
o (C) The 95% CI for the percentage of kissers who turn their heads to the right is
roughly 64.5% 4%.
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find more resources at oneclass.com
o (D) The 99.7% CI for the percentage of kissers how turn their heads to the right is roughly
64.5% 12%.
Confidence interval for a population mean: Computed as the sample mean a margin of error
(critical value corresponding to the middle XX% of the normal distribution times the standard
error of the sampling distribution).
o x̄ z * (s/rad(n))
o z* (the critical value) is approximately 2 for a 95% confidence interval and so on
Conditions for the confidence interval are the same for CLT conditions
o Independence: Sampled observations must be independent
▪ Can be verified by checking:
• Random sample/assignment
• If sampling without replacement, n < 10% of population
o Sample size/skew: n 30 if the population distribution is very skewed
▪ The more skewed the distribution, the larger the sample size needs to be
How do we find z* for other confidence intervals?
o When finding the specific z* for a 95% interval, we see that z* is 1.96 and -1.96
Part 2: (2) Accuracy v. Precision of Confidence Intervals
Confidence interval – suppose we took many samples and built a confidence interval from each
sample using the equation:
o Point estimate 1.96(SE)
o Then about 95% of these intervals would contain the true population mean ()
o Commonly used confidence intervals are 90%, 95%, 98%, and 99%
o Changing the confidence interval is simply changing the confidence interval
o As the confidence level increases, the interval gets wider
What drawbacks are associated with using a wider interval?
o A wider interval would be accurate, but it would not be as precise
o As the confidence level increases, the interval width becomes wider, and the accuracy
increases, but the precision decreases
How can we get high precision and high accuracy?
o Increase the sample size
▪ Will shrink our standard error and margin of error
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find more resources at oneclass.com
Document Summary
Part 1: (1) sampling variability and clt state and take the mean ages 18-24 at the 95% confidence level: we are 95% confident that 38. 1% to 43. 9% of the public believe that young adults, rather. Addresses sampling variability since we cannot conduct a study on the entire population. Ex: a study shows that the 41% of the public believes that young adults, rather than middle-aged. 49% of young adults at 18-34 have taken on a job for the sake of paying bills. Suppose we are interested in women"s heights in the us. We cannot calculate the entire population"s height means, so we take sample (cid:883),(cid:882)(cid:882)(cid:882) women in each. )f we don"t have access to sigma, which the population standard deviation, then we can use s, mean, and with a standard deviation equal to the population standard deviation divided by the square root of the sample size.
