Last lecture summary. 2345678 0 344434243533 Population 2015.

Last lecture summary

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Population 2015

Population 2014

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56 7

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průměr = 3.3

průměr = 3.0

Data 2015

Population:

4,3,3,5,0,4,4,4,3,4,2,6,8,2,4,3,5,7,3,3

25 samples (n=3) and their averages

3.3,5.3,3.6,4.3,2.3,3.0,3.6,3.0,5.3,5.6,3.3,4.3,3.3,4.0,5.6,4.3,4.3,4.6,6.3,3.3,4.0,3.3,4.6,3.0,4.3

http://blue-lover.blog.cz/1106/lentilky

2015, n = 3, number of samples = 25

2015, n = 3, number of samples = 50

2015, n = 3, number of samples = 300

2015, n = 3, all possible samples (1540)

2015, n = 5, all possible samples (42 504)

2015, n = 10, all possible samples (20 030 010)

Central limit theorem• Distribution of sample means is normal.

• The distribution of means will increasingly approximate a normal distribution as the sample size increases.

• Its mean is equal to population mean.

• Its standard deviation is equal to population standard deviation divided by the square root of .• is called standard error.

𝑆𝐸=𝜎 𝑥=𝜎√𝑛

𝑀 ¿𝜇𝑥=𝜇

ESTIMATION, CONFIDENCE INTERVALS

Statistical inference

If we can’t conduct a census, we collect data from the sample of a population.

Goal: make conclusions about that population

Demonstration• You sample 36 apples from your farm’s harvest of over

200 000 apples. The mean weight of the sample is 112 grams (with a 40 gram sample standard deviation).

• What is the probability that the mean weight of all 200 000 apples is within 100 and 124 grams?

What is the question?• We would like to know the probability that the population

mean is within 12 of the sample mean.

• But this is the same thing as

• But this is the same thing as

• So, if I am able to say how many standard deviations away from I am, I can use the Z-table to figure out the probability.

Slight complication• There is one caveat, can you see it?• We don’t know the standard deviation of a sampling

distribution (standard error). We only know it equals to , but is uknown.

• What we’re going to do is to estimate . Best thing we can do is to use sample standard deviation .

• . This is our best estimate of a standard error.• Now you finish the example. What is the probability that

population mean lies within 12 of the sample if the SE equals to 6.67?• 92.82%

This is neat!• You sample 36 apples from your farm’s harvest of over

200 000 apples. The mean weight of the sample is 112 grams (with a 40 gram sample standard deviation). What is the probability that the population mean weight of all 200 000 apples is within 100 and 124 grams?

• We started with very little information (we know just the sample statistics), but we can infere that

with the probability of 92.82% a population mean lies within 12 of our sample mean!

Point vs. interval estimate• You sample 36 apples from your farm’s harvest of over

200 000 apples. The mean weight of the sample is 112 grams (with a 40 gram sample standard deviation).

• Goal: estimate population mean

1. Population mean is estimated as sample mean. i.e. we say population mean equals to 112 g. This is called a point estimate (bodový odhad).

2. However, we can do better. We can estimate that our true population mean will lie with the 95% confidence within an interval of (interval estimate).

𝑥±1.96×𝑠

√𝑛

Confidence interval• This type of result is called a confidence interval

(interval spolehlivosti, konfidenční interval).

• The number of stadandard errors you want to add/subtract depends on the confidence level (e.g. 95%) (hladina spolehlivosti).

𝑥±𝑍×𝑠

√𝑛margin of error

možná odchylka

critical valuekritická hodnota

Confidence level• The desired level of confidence is set by the researcher,

not determined by data.• If you want to be 95% confident with your results, you add/subtract

1.96 standard errors (empirical rule says about 2 standard errors).• 95% interval spolehlivosti

Confidence level Z-value

80 1.28

90 1.64

95 1.96

98 2.33

99 2.58

80% 90%

95% 99%

1.28

1.96

1.64

2.58

Small sample size confidence intervals

• 7 patient’s blood pressure have been measured after having been given a new drug for 3 months. They had blood pressure increases of 1.5, 2.9, 0.9, 3.9, 3.2, 2.1 and 1.9. Construct a 95% confidence interval for the true expected blood pressure increase for all patients in a population.

• We will assume that our population distribution is normal, with and .

• We don’t know anything about this distribution but we have a sample. Let’s figure out everything you can figure out about this sample: • ,

• We estimate true population standard deviation with sample standard deviation

• However, we are estimating our standard deviation with of only seven! This is probably goint to be not so good estimate.

• In general, if this is considered a bad estimate.

William Sealy Gosset aka Student• 1876-1937• an employee of Guinness

brewery• 1908 papers addressed the

brewer's concern with small samples• "The probable error of a mean".

Biometrika 6 (1): 1–25. March 1908.• Probable error of a correlation

coefficient". Biometrika 6 (2/3): 302–310. September 1908.

Student t-distribution• Instead of assuming a sampling distribution is normal we

will use a Student t-distribution.• It gives a better estimate of your confidence interval if you

have a small sample size.• It looks very similar to a normal distribution, but it has

fatter tails to indicate the higher frequency of outliers which come with a small data set.

Student t-distribution

Student t-distribution

df – degree of freedom (stupeň volnosti)

Back to our case

• Because sample size is small, the sampling distribution of the mean won’t be normal. Instead, it will have a Student t-distribution with .

• Construct a 95% confidence interval, please

for𝑛<30 :𝑥 ±𝑡𝑛−1×𝑠

√𝑛

• Just to summarize, the margin of error depends on1. the confidence level (common is 95%)

2. the sample size • as the sample size increases, the margin of error decreases• For the bigger sample we have a smaller interval for which we’re

pretty sure the true population lies.

3. the variability of the data (i.e. on σ)• more variability increases the margin of error

• Margin of error does not measure anything else than chance variation.

• It doesn’t measure any bias or errors that happen during the proces.

• It does not tell anything about the correctness of your data!!!

neco×𝑠

√𝑛