PROBABILITY AND STATISTICS

WEEK 9-10

Onur Doğan

The sampling distribution of the sample statistics

Onur Doğan

The sampling distribution of the sample statistics

Consider a population of N elements from which we can obtain the following distinct data: {0, 2, 4, 6, 8}.

•Form samples of size 2 for this population.

•Define their means and figure the bar chart of the means.

•Define the sampling distribution of the sample ranges and figure bar chart.

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The Central Limit Theorem

The mean is the most commonly used sample statistic and thus it is very important. The central limit theorem is about the sampling distribution of sample means of random samples of size n.Let us establish what we are interested in when studying this distribution:

1) Where is the center?2) How wide is the dispersion?3) What are the characteristics of the distribution?

The central limit theorem gives us an answer to all these questions.

Onur Doğan

The Central Limit Theorem

Let µ be the mean and σ the standard deviation of a population variable. If we consider all possible random sample of size n taken from this population, the sampling distribution of sample means will have the following properties:

c) if the population is normally distributed the sampling distribution of the sample means is normal; if the population is not normally distributed, the sampling distribution of the sample means is approximately normal for samples of size 30 or more. The approximation to the normal distribution improves with samples of larger size.

Onur Doğan

The Central Limit Theorem

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The Central Limit Theorem

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The Central Limit Theorem

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Example

Consider a normal population with µ=100 and σ=25. If we choose a random sample of size n = 36, what is the probability that the mean value of this sample is between 90 and 110? In other words, what is P(90 < x < 110)?

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Example

The average male drinks 2L of water when active outdoor s(with standard deviation of 0,7 L). You are planning a full day nature trip for 50 men and bring 110 L of water. What is the probability that you will run out?

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Confidence Intervals

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Confidence Interval on the Mean of a Normal Distribution, Variance Known

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Confidence Interval Formula

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Example

Suppose that the life length of a light bulb (X; unit: hour) follows the normal distribution N(y, 402). A random sample of n = 30 bulbs is tested and the sample mean is found to be 780 hours.

•Construct a 95% two sided confidence interval on the mean life length (µ) of a light bulb.•Find a sample size n to construct a two-sided confidence interval on µ with an error = 20 hours from the true mean life length.(use α= 0.05)

Onur Doğan

Example

• The diameter of a hole (X; unit: in.) for a cable harness is normal with σ2= 0.012. A random sample of n = 10 yields an average diameter of 1,5045 in.

• Construct a 99% upper-confidence bound on the mean diameter (p) of the hole.

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Confidence Interval on the Mean of a Normal Distribution,Variance Unknown (t distribution)

Small Sample CI?

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Example

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 standart deviation). What is the probability that the mean weight of all 200 000 apples is within 100 and 124 grams?

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A Large-Sample Confidence Interval for a Population Proportion

Confidence Interval Formula:

Sample Size Selection

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Example

A sample of n = 40 bridges in a city is tested for metal corrosion, and x = 28 bridges are found corroded.

•Construct a 95% two-sided confidence interval on the proportion of corroded bridges (p) in the county.

•Determine a sample size n to establish a 95% confidence interval on p with an error = 0.05 from the true proportion.

Onur Doğan

Example

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In a teaching district school management allows theacher using omputer in their lessons. From the 6000 teachers in district, 250 were randomly selected and asked if they fekt that computers were an essential teaching tool for their calssrom. Of those selected, 142 teachers felt that computers were an essential teachin tool.

•Calculate a 99% confidence interal for the proportion of teachers who felt that computers are an essential teaching tool

Summary for Confidence Intervals

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Hypothesis Testing

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Test Regions

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Test Errors

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Hypothesis Testing Procedure

Tests on the Mean of a Normal Distribution, Variance Known

• .

Example

For several years, a teacher has recorded his students' grades, and the mean, µ for all these students' grades is 72 and the standard deviation is σ = 12. The current class of 36 students has an average x = 75,2 (higher than µ = 72) and the teacher claims that this class is superior to his previous ones. Test the teacher’s claim for the level of significance = α=0,05.

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Relationship Between Hypothesis Test, CI and p-value

Tests on a Population Proportion

• .

Example

For the bridge example; a specialist claims that more than half of the

bridges gave been corroded in the city. Test the specialist’s claim

with %95 confidence.

Example

• Suppose that a factory is producing wheels for airbuses. The manufacturer claims that they produce wheels 3 meters diameter.

• The quality control department of the buyer firm investigate a sample from the daily product. They controlled 36 wheels and found that average diamater is 2,92 and s.d. is 0,18.

• Test the claim at α=0.05 significance level

Example

• According to a recent poll 53% of Americans would vote for the incumbent president. If a random sample of 100 people results in 45% who would vote for the incumbent, test the claim that the actual percentage is 53%. Use a 0.10 significance level.

Example

• The national weather service says that the mean daily high temperature for July in İzmir is 42°C. A local weather service wants to test the claim of 42°C because it believes it is different. A sample of mean daily high temperatures for October over the past 31 years yields =44°F and s=3.8°C. Test the claim at α=0.01 significance level.

Example

• In a clinical study of an allergy drug, 108 of the 203 subjects reported experiencing significant relief from their symptoms. At the 0.01 significance level, test the claim that more than half of all those using the drug experience relief.