Chapter 9

1. Chapter 9 Indentify and describe sampling distributions.

2. Vocabulary • Parameter • A number that describes the population (p) • Statistic • A number that describes a sample. A statistic is used to estimate an unknown parameter. ( )

3. Mean of population is μ • Mean of sample is

4. Pg 489 #1 a. state whether it is a parameter or a statistic and b. use appropriate notation to describe each number; for ex p = 0.65 • A carload lot of ball bearings has mean diameter 2.5003 centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. By chance, an inspector chooses 100 bearings from the lot that have mean diameter 2.5009 cm. Because this is outside the specified limits, the lot is mistakenly rejected.

5. Can the statistic be used to represent the parameter? • Sampling Variability – the value of the statistic varies in repeated random sampling • Sampling Distribution – the distribution of values taken by the statistic in all possible samples of the same size from the same population.

6. The distribution of the sample proportion p hat from SRSs of size 100 drawn from a pop with p = .7. The results of 1000 samples taken.

7. Describing sampling distributions • Overall shape • Center of distribution • Spread (variability) • Outliers

8. Proportion of sample who watched Survivor II in samples of n = 100 Overall shape of distribution is symmetric and approx normal Center – very close to the true value p=.37, in fact the mean of the 1000 ‘p hat’s is .372 and their median is exactly .37 Values of ‘p hat’ have a large spread. Range from .22 to .54, stand. dev is .05 There are no outliers or other important deviations from the overall pattern.

9. The approx sampling dist. of the sample from SRSs of size 1000 On the same scale as the SRSs of size 100 from previous slide

10. N = 1000, redrawn with expanded scale to better display shape. Center is again close to .37, mean is .3697, and median is .37 Spread, is much less from .321 to .421, stand dev is .016 Shape again close to normal.

11. Unbiased Statistic – a statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. • Variability of a Statistic – described by the spread of its sampling distribution. • This spread is determined by the sampling design and size of the sample. • Larger samples size smaller spreads.

12. Rule of Thumb • Population at least 10 times as large as sample. The spread of the sampling distribution is approximately the same for any population size. • High/Low Bias • High/Low Variability

14. 2,3,4,10,11,17 • In class 1 & 8

15. Section 9.2 – Sampling Proportions • Sampling Distribution of a Sample Proportion – choose a SRS of size n from a large population with population parameter p, have some characteristic of interest. Let be the proportion of the sample having that characteristic. Then • The mean of the sampling distribution is exactly p so…… • The standard deviation of the sampling distribution is • is less variable with larger samples • decreases as n increases.

16. Rule of Thumb 1 • Use the recipe for the standard deviation of only when the population is at least 10 times as large as the sample

17. Rule of Thumb 2 • We will use the normal approximation to the sampling distribution of for values of n and p that satisfy np ≥10, and n(1-p)≥10

18. Example 9.7 applying to college SRS of size n=1500 • SRS of 1500 first year college students whether they applied for admission to any other college. • 35% of all first – year students applied to colleges besides the one they are attending. • What is probability that random sample of 1500 students will give a result within 2 percentage points of this true value? Population has proportion p=.35, so sampling distribution of ‘ ’ has mean So standard deviation? From first Rule of Thumb, pop must contain at least 10(1500) = 15,000 people. There are 1.7 million first-yr college students so…

19. Can we use normal distribution to approximate the sampling distribution of ‘p hat’? Second rule of thumb: np=1500(.35)=525, n(1-p)= 1500(.65) = 975 Both are MUCH larger than 10 so YES, normal approx is accurate.

20. What is probability that ‘p hat’ falls between .33 and .37? Area of shaded region is 0.33≤’p hat’≤0.37

23. 20, 21, 22, 25, 27, 30 • In class 19

24. 9.3 Sample Means • Mean and Standard deviation of a sample mean suppose that is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. • Then the mean of the sampling distribution of is • The standard deviation is

25. The sample mean ‘x bar’ is an unbiased estimator of the population mean μ. • The values of ‘x bar’ are less spread out for larger samples • Their standard deviation decreases at the rate • Use when ROT#1 is met.

26. Sampling Distribution of a Sample Mean from a Normal Population • Draw a SRS of size n from a population that has the normal distribution with mean μ and standard deviation σ. Then the sample mean ‘x bar’ has the normal distribution with mean μ and standard deviation

27. In class 31, 35, • Homework 32, 33, 34, 36, 39, 40, 42