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).
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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 = ?
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:
1) The interval between 452 and 470 contains the true population mean
2) Our SRS was one of the few samples for which x-bar was NOT within 9 points of the true population mean.
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.