Stats 6.1-3

Unbiased estimator

A statistic used to estimate a parameter if the mean of its sampling distribution is equal to the value of the parameter being estimated.

Interpret the standard deviation

If ___ took many samples of size n, the number of ___ would typically vary by about ___ from the mean of __.

There is one dot on the graph at p̂ = .38 . Explain what this dot represents.

In one SRS of size n, 38% of __ were ___.

The Large Counts condition

Suppose X is the number of successes in a random sample of size n from a population with proportion of successes p. The Large Counts condition says that the distribution of X will be approximately normal when np ≥ 10 and n(1 - p) // nq ≥ 10

Decreasing sampling variability

The sampling distribution of any statistic will have less variability when the sample size is larger.

What would happen to the sampling distribution of the sample mean if the sample size were n = 50 instead? Justify. What is the practical consequence of this change in sample size?

The sampling distribution of the sample mean will be more variable because the sample size is smaller. The estimated mean ___ will typically be farther away from the true mean ___. In other words, the estimate will be less precise.

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sampling distribution of the sample count X

describes the distribution of values taken by the sample count X in all possible samples of the same size from the same population.

sampling distribution of a statistic

is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

Unbiased estimator:

mean equals the value of the population mean

Bias

means that our aim is off and we consistently miss the bullseye in the same direction. That is, our sample values do not center on the population value.

High Variability

means that repeated shots are widely scattered on the target. In other words, repeated samples do not give very similar results.

Sample Statistics

x̄ (the sample mean) p̂ (the sample proportion) s (the sample SD)

x̄ (the sample mean) estimates...

μ (the population mean)

Population Parameter

μ (the population mean) p (the population proportion) σ (the population SD)

Mean of the sampling distribution of X

μx = np.

s (the sample SD) estimates...

σ (the population SD)

The standard deviation of the sampling distribution of X

σ = √np(1-p) or √npq

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Practical consequence of increasing sample size:

More reliable and precise - closer to the true mean

A Pew Research Center poll asked 1102 12- to 17-year-olds in the United States if they have a cell phone. Of the respondents, 71% said "Yes."

Population: all 12- to 17-year-olds in the United States. Parameter: p = the proportion of all 12- to 17-year-olds with cell phones. Sample: the 1,102 12- to 17-year-olds contacted. Statistic: the sample proportion with a cell phone, p̂ = 0.71.

Tom is roasting a large turkey breast for a holiday meal. He wants to be sure that the turkey is safe to eat, which requires a minimum internal temperature of 165°F. Tom uses a thermometer to measure the temperature of the turkey breast at four randomly chosen points. The minimum reading he gets is 170°F.

Population: all possible locations in the turkey breast. Parameter: the true minimum temperature in all possible locations. Sample: the four randomly chosen locations. Statistic: the sample minimum, 170°F.

When not app to use normal model:

not np ≥ 10 and n(1 - p) // nq ≥ 10 It is not appropriate to use a normal distribution model because np is not greater than 10.

p̂ (the sample proportion) estimates...

p (the population proportion)