Switch on the television or open your favorite news website; every few weeks you will find people’s opinion about something. Here’s one that came out a few days back. Our best days are ahead – so they say.
If you are a regular reader here, you would have already asked about the survey method and sample size.
Results for this Gallup poll are based on telephone interviews conducted June 18-24, 2018, with a random sample of 1,505 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±3 percentage points at the 95% confidence level.
The margin of sampling error is ±3% points at the 95% confidence level.
“Margin of sampling error,” “95% confidence level” – Have you ever asked yourself what these are?
In another news, the state of NJ passed a new “multi-millionaires tax,” Airbnb tax and Uber/Lyft tax among others. I hope the residents of NJ were asked about some of these increases, and this is what they wanted.
If they were to be asked, how do we know how many people to survey?
If a subset of people were to represent the population of NJ, how do we know that the resulting opinion is what the entire population actually wants?
Jill is having fun this summer. He is one of the volunteers at the BIG CITY FISHING SUNDAYS on Pier 84 . He is busy collecting water samples from the Hudson River. This week he computed the mean dissolved oxygen level as 4.9 mg/L. If the acceptable water quality level is a mean dissolved oxygen of 5 mg/L, how reliable is the estimate that Jill got? Can it be deemed acceptable?
In another news, Wilfred Quality Jr. is still thinking about his grandfather’s question: “how confident are you that the true value lies within two standard deviations?”
Jill’s boss is a variance control freak. Can we give him an upper bound on how much the dissolved oxygen in the Hudson River can vary? He usually likes to be 99% confident.
In another news, St Patrick’s Day should remind you of all those pints of Guinness, and “Student.” 
No, I am not under the influence.
All these questions can be answered with a little knowledge about how to describe uncertainty in estimates.
To take you back to Lesson 62, the objective of inference is to use a ‘sample’ to compute a number which represents a good guess for the true value of the parameter. The true population parameter is unknown, and what we estimate from the sample will enable us to obtain the closest answer in some sense.
The cliched way of saying this is that the mean 
and variance 
of the sample data are good guesses (estimates or estimators) of the mean 
and variance 
of the population.
A different sample will yield a different estimate for the parameter. So we have to think of the estimate as a range of values or a probability distribution instead of a single value.
For any estimator 
, we can compute the expected value ![E[\hat{\theta}]](https://i0.wp.com/www.dataanalysisclassroom.com/wp-content/ql-cache/quicklatex.com-f028d1740cfd78bd90ac5b8f18f3db7b_l3.png?resize=31%2C22&ssl=1 "Rendered by QuickLaTeX.com")
as a measure of the central tendency, and ![V[\hat{\theta}]](https://i0.wp.com/www.dataanalysisclassroom.com/wp-content/ql-cache/quicklatex.com-8e2a80a47d81a78d1fc1a0cdff9c9906_l3.png?resize=31%2C22&ssl=1 "Rendered by QuickLaTeX.com")
as a measure of the spread of the distribution giving us a range of values. Like this.
This range is called an interval. Naturally, the interval will be wider for data that has more variability. We can be confident that the truth may be in this interval if we have good representative samples.
How confident we are is a probability statement. For example, if there is a 95% probability that the interval contains the true value, it is equivalent to saying that this interval is the 95% confidence interval.
The interval will have a stated probability of containing the truth, the degree of plausibility specified by the confidence level → 95% interval or 99% interval.
Over the next several lessons, we will dive into the core concepts of confidence intervals, how to construct them for different population parameters and use them for designing experiments.
Meanwhile, do you know about Guinness and Student? Guinness, I am sure, but are you confident that you know Student?
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