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10.1: Estimating with Confidence

- Statistical inference uses language of probability to express the strength of our conclusions
- 2 types of formal statistical inference: confidence intervals (Ch. 10) and significance tests which assesses evidence for a claim about a population (Ch. 11)

Suppose you want to estimate the mean SAT Math score for the more than 350,000 high school seniors in CA. Only about 49% of these students take the SAT. These self-selected seniors are planning to attend college and so are not representative of all CA seniors. You know better than to make inferences about the population based on any sample data. At considerable effort and expense, you give the test to an SRS of 500 CA high school seniors. The mean for your sample is 461. If the standard deviation of SAT Math scores is 100, what can you say about the mean score in the population of all 350,000 seniors?

What we know about

- …it is approximately….?
- We want a better estimate of x-bar.
- What we need to ask is, “How would the sample mean vary if we took many samples of 500 seniors from this population?
- 3 facts about x-bar

1) CLT: Shape of the mean of 500 scores has a distribution that is = ?

2) Mean of this sampling distribution = ?

3) Standard deviation of x-bar for an SRS of 500 = ?

Take many samples of n=500 with mean SAT math score for each sample

- If we collect all these sample means and display their distribution, we get the normal distribution with mean = and standard deviation = 4.5.

1) To estimate mew, use x-bar of our random sample.

2) Although x-bar is an unbiased estimate of mew, it will rarely be exactly equal of mew, so our estimate has some error.

3) In repeated samples, the values of x-bar follow (approximately) a normal distribution.

4) The 68-95-99.7 rule says:

5) Whenever x-bar is within 9 points of mew, mew is also within 9 points of x-bar (in 95% of samples)

6) Therefore, in 95% of all samples:

Steps to statistical estimation2 possibilities

1) The interval between 452 and 470 contains the true population mean

or

2) Our SRS was one of the few samples for which x-bar was NOT within 9 points of the true population mean.

Users usually pick C > 90% because they want to be sure of their conclusions.

- A confidence interval has 2 Parts:
- An interval calculated from the data
- A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples.

Top: Result of drawing many SRS’s from the same population and calculating a 95% confidence interval from each sample.

- Sampling distribution of x-bar shows long-term pattern of this variation.
- 25 SRS’s, 95% Confidence

Example 2

- To find an 80% confidence interval, we must catch the central 80% of the normal sampling distribution of x-bar. In catching the central 80%, we leave out 20%, or 10% in each tail.

Recall: The sampling distribution of x-bar is normal if the population is normal.

If the population isn’t normal and n is sufficiently large, the sampling distribution is stillapproximately normal (CLT: N>30).

*

A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Too much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in milllivolts. Some variation ins inherent in the production process. Careful study has show that when the process is operating properly, the standard deviation of the tension readings is 43 mV. Here are the tension readings from an SRS of 20 screens from a single day’s production.

269.5 297 269.6 283.3 304.8 280.4 233.5 257.4 317.5 327.4

264.7 307.7 310 343.3 328.1 342.6 338.8 340.1 374.6 336.1

Construct a 90% confidence interval for the mean tension of all the screens produced on this day.

How confidence intervals behave

- Larger samples give smaller margins of error.
- The margin of error gets smaller…
- When z* gets smaller. Smaller z* is the same as smaller confidence level.
- Population standard deviation gets smaller.
- N gets larger.

Choosing the Sample Size

- To obtain a desired margin of error (m), substitute the value of z* for your desired confidence level, and solve the inequality for n.
- Ex: Company management wants a report of the mean screen tension for the day’s production accurate to within 5 mV with 95% confidence. How large a sample of video monitors must be measured to comply with this request?

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