The t-distribution

Question 1

Why can’t we use the sample standard deviation of the sample to work out the standard error instead of the standard deviation of the parent population?

Answer 1

Because it doesn’t closely follow the standard normal distribution (SDN) unless the sample size is very large

Question 2

What is the student’s t?

Answer 2

A family of distributions. There are many of them for different parameters

Question 3

What are the similarities of the student’s t with the SND?

Answer 3

• bell shaped
• symmetric
• uni-modal (one mode)

Question 4

What are the differences between t-distributions and SND?

Answer 4

• T-distributions have a lower peak than SND and higher tails
• Has higher tails because the area under the peak is always 1. If peak goes down, then tails go up.
• Have more variance (more spread out) than SND
• More weight in the tails

Question 5

When do you use the t-distribution?

Answer 5

instead of SND when parent pop s.d. isn’t known

Question 6

How do you work out your t score?

Answer 6

Tm = (m-u)/estimated standard error (e.s.e)

Question 7

How do you work out your estimated standard error?

Answer 7

Substitute the the standard deviation of the parent population for the standard deviation of the sample population

Question 8

How do you work out which parameter your t-distribution follows?

Answer 8

N-1

Question 9

How do you read the critical values table?

Answer 9

• work out your parameter and find this on the right hand side
• choose whether you need to look at one tailed or two tailed hypothesis
• find the t-value that corresponds with what you are looking for

Question 10

If we have a sample of 6 people from a normal population. What is the probability that t = (m-u)/e.s.e will be greater than 3.0 (i.e. more than 3 e.s.e’s above the pop mean)?

Answer 10

• Using the table we can’t say for sure but we can say that:
• P< 0.025
• P > 0.01

Question 11

What do we need to know in order to be able to work out the 95% confidence interval for a population mean when the population standard deviation is unknown?

Answer 11

As long as we can work out the value c- value that identifies 2.5% area above or below the tail of the t-distribution then we can work out u-(c x s/square root n)

Question 12

What can you say for a sample of size N drawn at random from a normal population N(u, sigma) with unknown s.d.; sample mean m & s.d. s?

Answer 12

The 95% CI for the population mean is centred on the sample mean m and goes from m-(c x s/square root N) to m+(c x s/ square root N)

Question 13

What is the 95% confidence interval for m with: Sample (N=6) m = 7.33, sample s.d. (s) = 3.78, e.s.e = 1.54

Answer 13

• The 95% CI for the population mean is centred on sample mean m
• And goes from 7.33 – (c x 1.54) to 7.33 + (c x 1.54)
• So we have t-distribution v= 5
• We are interested in the column with the one-tailed hypothesis 0.025 or two-tailed hypothesis 0.05
• T-value – 2.571
• Therefore 95% of the area between -2.571 and +2.571
• (typically t-values will be more than 1.96 because you have to go further out to isolate an equivalent area because the t-distribution has higher tails)
• Therefore: 7.33 – (2.571 x 1.54) to 7.33 + (2.571 x 1.54)
• So C.I. lower = 7.33 – (2.571 x 1.54) = 3.37
• C.I. upper = 7.33 + (2.571 x 1.54) = 11.30
• 95% CI = (3.37, 11.30)

Question 14

When do we use the 1 sample t-test?

Answer 14

in any situation we would use the z-test but the population standard deviation is unknown

Question 15

How would you conduct this 1 sample t-test: Visiting Aliens from Ziltodia 10 are concerned they are shorter on earth due to there being more gravity here than back home. Population mean height on Ziltodia 10 is known to follow a normal distribution with mean 24.5cm, but the s.d. in the population is unknown.

Sample statistics:

• N = 36
• M = 23.5
• S (sample s.d.) = 2.4
• U = 24.5

Answer 15

• Step 1 – Hypotheses:
- Null hypothesis (H0):
 Ziltodians on earth will not be shorter than on ziltodia 10
 (H0): the sample mean came from a population with mean uh = 24.5)
- Research Hypothesis (H1):
 Ziltodians on earth will be shorter than on Ziltodia 10
 (H1): the sample mean came from a population with mean uh < 24.5)
• Step 2 – Data
- Design experiment and collect data
- We get a random sample of visiting Ziltodians (N=36) and measure their height
- N = 36
- M = 23.5
- S (sample s.d.) = 2.4
- U = 24.5
- Sigma = ?
• Step 3 – evaluate inconsistency with H0
- How inconsistent are the data with H0?
- Assume H0 is true. What is the conditional probability of having got a mean as low/ lower than ours?
- Can’t covert our sample mean to z-score because s.e. unknown
- E.s.e = 2.4/square root 36 = 0.4
- t = (23.5=24.5)/ e.s.e. = -2.5
- Follows t distribution t(35)
- Interested in area to the left of our t-statistics
- Using the table we can’t say for sure but we can say if p< 0.05
- So p<0.05
• Step 4 – Reject/ fail to reject
- Based on step 3 we now know that the conditional probability of having got a sample mean as low or lower than the one we got if we assume H0 is true is less than 0.05
- P <0.05
- Because p<0.05 (5%) we can reject the null hypothesis in favour of the research hypothesis – it is just too unlikely that we would have got a mean this much below the population mean if H0 were true
• Step 5
- Interpret
- We reject the null (H0) so we can say that we have evidence for the research hypothesis (H1) and that our sample came from a population with mean uh<24.5
- We have evidence that Ziltoidans are shorter on earth than ziltodia