STAT1008 Study Guide - Final Guide: Central Limit Theorem, Confidence Interval, Statistical Hypothesis Testing

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17 May 2018

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Distribution of a Sample Proportion

• The standard error for  



• The larger the sample size, the smaller the SE.

Sufficiently Large n

• The larger the sample size, the more like a normal distribution it becomes. A normal

distribution is a good approximation as long as np10 and n(1 - p) 10.

Central Limit Theorem for 



•  ~ N (



• A normal distribution is a good approximation as long as np 10 and n(1-p) 10.

6.2 Confidence Interval For a Single Proportion

• When doing inference, we don’t know p so substitute p̂ as it is our best guess.

• Provided the sample size is large enough so that np̂ ≥  and n(1-p̂≥ , a confidence interval

can be computed by *  



Margin of Error

• If we want to estimate a population proportion to within a desired ME, we should select a

sample of size

• Neither p or p̂ is known in advance. To be conservative, use p = 0.5.

• N ≈ (1/ME)2

6.3 Test For a Single Proportion

Hypothesis Testing

• For hypothesis testing, we want the distribution of the sample proportion assuming H0 is true.

• To test Ho: p=po:

Test for a Single Proportion

•

• If npo  10 and n(1-po) 10 , then the p-value can be computed as the area in the tail(s) of a

standard normal beyond z.

6.4 Distribution of a Sample Mean

Standard Error of Sample Means

• The SE for  = 



• The larger the sample size (n), the smaller the SE.

Central Limit Theorem for Sample Means

• If n is sufficiently large:

• A normal distribution is usually a good approximation, as long as n30.

The Distribution of Sample Means Using the Sample Standard Deviation

• Usuall, e do’t ko the populatio SD



, so estimate it with the sample SD, s.

n=z* /ME

( )

p(1-p)

) / n

z=p-p0

-p0

) /

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Document Summary

Sufficiently large n: the larger the sample size, the smaller the se, the larger the sample size, the more like a normal distribution it becomes. A normal: the standard error for is (cid:4666)(cid:2869) (cid:4667) distribution is a good approximation as long as np 10 and n(1 - p) 10. Central limit theorem for : ~ n (, (cid:4666)(cid:2869) (cid:4667) (cid:4667, a normal distribution is a good approximation as long as np 10 and n(1-p) 10. can be computed by * (cid:4666)(cid:2869) (cid:4667) If we want to estimate a population proportion to within a desired me, we should select a sample of size n = z * /me. )2 p(1- p: neither p or p is known in advance. For hypothesis testing, we want the distribution of the sample proportion assuming h0 is true: to test ho: p=po: Test for a single proportion p - p0 z = p0 (1- p0 ) / n standard normal beyond z.

Related Questions

6. As the test statistic becomes larger, the p-value

(a) gets smaller

(b) becomes larger

(d) becomes negative.

7. To describe the sampling distribution of the sample proportion, we use

(a) The sample proportion

(b) The population proportion

(d) The population mean

8. When small samples are used to estimate a population mean, in cases where the population standard deviation is unknown and the original distribution is normal:

(a) There always will be a large amount of sampling error.

(b) the t-distribution must be used to obtain the critical value.

(d) the z distribution must be used to obtain the critical value.

periwinkleyellowjacket100

Never forget that even small effects can be statistically significant if the samples are large. During a three-year period, 8 of the 57 headed by men and 6 of the 35 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the p-value for the hypothesis that the same proportion of women's and men's businesses fail. (Use two-sided alternative). What can we conclude (use alpha as 0.05)?

The p-value was ___________ so we conclude that ____________.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is 180 out of 1050 businesses headed by women and 240 out of 1710 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?

The p-value was ___________ so we conclude that ____________.

c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a p-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses that fail for the settings of both (a) and (b).

Interval for smaller samples: ___________ to ___________.

Interval for larger samples: ___________ to ___________.

What is the effect of larger samples on the confidence interval?

viridiancoyote964

STAT1008 Study Guide - Final Guide: Central Limit Theorem, Confidence Interval, Statistical Hypothesis Testing

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