bio stats ch 11

Sampling distribution of the mean

Instead of looking at the values of individual observations, we look at the average of each sample

Steps conceptually:

• Take a sample from the population.

• Calculate the sample mean.

• Repeat this many times.

• Plot all those sample means.

• The result is the distribution of sample means.

Important properties

• The mean of this distribution equals the population mean (μ).

• The standard deviation is called the standard error:

Example of sampling distribution of the mean

Suppose a population has heights.

• You take many samples of n = 30 people:

• Sample 1 → mean = 170 cm

• Sample 2 → mean = 168 cm

• Sample 3 → mean = 172 cm

• Sample 4 → mean = 169 cm

• If you plot all these means, you get the distribution of sample means.

Distribution of sample means additional equations

• Recall that Y has a sampling distribution that is normal

• Z standardization for distribution of sample means:

Similar formula but we are working with standard error now

• Distribution changes based on sample size

• Good for small sample size

• Normal distribution good for large sample size (inc in sample size makes it narrower)

• This allows you to use t values and get accurate conclusions based on small sample size

Student’s t distribution (Gosset)

• Useful conclusions from small-scale studies

• made statistics a useful way for Guinness to increase its profits

• tell which products were better based on a few samples

• didn’t want other companies to steal it so he had to publish it anonymously

• similar to standard normal distribution (Z) but with fatter tails

• the t-distribution here has df=4

• as the sample size increases the t distribution becomes more like the standard normal distribution

• always work with 2-sided

• as critical value like Z

• get values with table

Assumptions of one-sample t test

Data are a random sample from the population

• Variable is normally distributed in the population

  • Few variables in biology are exact match to normality

  • But in many cases the test is robust to departures from normality (more on this later)

When to use Z or T test

Use a Z-test when:

✔ The population standard deviation (σ) is known

✔ Sample size is usually large (n ≥ 30) (not strictly required if σ is known)

✔Data is approximately normal (or large n → CLT helps)

• central limit theorem

Use a One-sample t-test when:

✔The population standard deviation (σ) is unknown

✔You use the sample standard deviation (s) instead

✔Especially important when n is small (< 30)

Why t instead of z?

Because when you don’t know σ, there is extra uncertainty, and the t distribution accounts for that (wider tails).

can use the t-distribution to accurately calculate a confidence interval for the mean of a population with a normal distribution

Example: Stalk-eyed flies

1) what is the confidence interval?

if mean is inside confidence interval you know the CI is right

df = 8

Example

1) Value from the table

2) Standard error

3) CI

4) Is the mean inside the confidence interval?

df = 5

2) SE

3) CI

4) Yes

Example

1) Value from the table

2) Standard error

3) CI

4) Is the mean inside the confidence interval?

1) Df = 4

2) SE

3) CI

Example

1) Value from the table

2) Standard error

3) CI

4) Is the mean inside the confidence interval?

df = 3

2)

3)

One sample T-test

the one sample t-test compares the mean of a random sample from a normal population with the population mean proposed in a null hypothesis

compute p-value: probability of this t-statistic (or more extreme) given the null hypothesis (u=98.6) is true

even with t test as sample size increases your data is more certain and you can have more certainty

Example: Human body temp

normal: 98.6 degrees F

• is this really normal?

• Sample of 25 healthy people

One sample t-test

• H0: mean body temp = 98.6

• HA: mean body temp does not equal 98.6

1) Standard error

2) t and Df

3)P-value —> H0

4) How much uncertainty is there in our estimate of u

1)

2)

3) compute p-value: probability of this t-statistic (or more extreme) given the null hypothesis (u=98.6) is true

Fail to reject the null hypothesis because the t-value is inside the confidence interval range

• So the data is consistent with the true H0

4)

Example: Sleep duration

Recommended sleep duration = 8 hours

• Is this really the case?

• A researcher collects a sample of 20 college students and records the number of hours they sleep per night.

• One-sample t-test

  • H0: mean sleep duration = 8 hours

  • HA: mean sleep duration ≠ 8 hours

1) Standard error

2) t and Df

3)P-value —> H0

4) How much uncertainty is there in our estimate of u (CI)

1)

2)

3)

Decision rule: Reject H0 if ∣t∣ > 2.09:

Since 3.51>2.09

The test statistic falls outside the 95% central region. Because the statistic falls in the rejection region, we reject H0.

Interpretation:

There is statistically significant evidence that the average sleep duration of college students is different from 8 hours

4)

because 8 hours is not contained in the 95% CI we reject H0

Estimating other statistics (spread of the sample distribution —> standard deviation or variance)

df = 19

Example: Birth weight

Average birth weight in the population = 3400 g

• Is the average birth weight in this hospital the same?

• A sample of 30 newborns is collected. (small so t-test)

• One-sample t-test

  • H0: mean birth weight = 3400g

  • HA: mean birth weight ≠ 3400g

1) calculate SE

2) calculate t value

3) calculate df and compare with table

4) calculate CI

5) decision

6) CI for variance

2)

3)

4)

5)

Value falls just beyond cutoff

Association really weak because you are very close to the borderline

Because you are slightly outside you must reject. DON'T ROUND TOO EARLY

6)

df= n-1 = 30-1 = 29

Example: cholesterol

Average cholesterol level in the population = 200 mg/dL

• Is the average cholesterol level in patients at this clinic the same?

• A sample of 25 patients is collected.

• One-sample t-test

  • H0: mean cholesterol level = 200 mg/dL

  • HA: mean cholesterol level ≠ 200 mg/dL

1) what is standard error

2) what is the t value

3) df and table value

4) CI

5) decision

1)

2) what is the t value

3) df

4) CI

5)