Estimation and significance tests and p values

Question 1

What are sampling errors?

Answer 1

Samples provide an incomplete picture of the population.
✓ Sample estimates (e.g.:means), calculated from multiple samples from the same population, will then have a distribution of differing values that is known as the ‘sampling distribution’.
✓ Different samples will give different estimates- called ‘sampling error’.

Question 2

What is standard error?
Formula?

Answer 2

✓A standard error (SE) is an indication of the extent of the sampling error
✓For a sample mean it can be calculated from: standard deviation divided by the square root
of the sample size SE= SD / 𝑁

Question 3

What is the important factors to a pricise result?

Answer 3

A: Bigger sample size, estimate closer to true mean
B: Smaller spread of data (standard deviation), estimate closer to true mean

Question 4

What does the standard error?

Answer 4

✓Standard error tells us how much a sample mean tends to vary from the population mean (true mean). It provides an estimate of the precision of the sample mean.

Question 5

What do SE?
What happens if you change the SD?

Answer 5

SE= SD/ 𝑁
* Changing SD or N increases or decreases the precision.
* Smaller variation (SD) or larger sample size (N) ֜——> smaller SE֜ —-> More precise estimate

Question 6

Assumptions in calculating the confidence interval?

Answer 6

✓Normal data or large sample
✓the sample is chosen at random from the
population
✓the observations are independent of each other
✓the sample is not small (at least 60)

Question 7

How do u calculate the standard error?

Answer 7

Standard error of mean (standard deviation divided by square root of the sample size) = 10.1/45.9 = 0.22

Question 8

If the mean is 0.22 what is 95% confident?

Answer 8

95% confident that true mean is in the range: 15.6 – 1.96 x 0.22 to 15.6 + 1.96 x 0.22

Question 9

Assumptions for the true mean?

Answer 9

✓the sample is chosen at random from the population
✓the observations are independent of each other
✓the proportion with the characteristic is not close to 0 or 1
✓np and n(1-p) are each greater than 5 (large sample)

Question 10

How to calculate the standard error?

Question 11

Define confidence interval and proportion of it?

Answer 11

✓ A confidence interval is a range in which we expect the true population value to lie

for the mean from a large sample the 95% confidence interval is:
sample mean -1.96 standard errors to
sample mean + 1.96 standard errors

for a proportion the 95% confidence interval is: sample proportion -1.96 standard errors
to
sample proportion + 1.96 standard errors

Question 12

What is the aim of hypothesis testing?

Answer 12

Hypothesis testing is a procedure used to:
✓evaluate the strength of evidence from a
sample
✓and to assess how reliably one can extrapolate findings in a sample to the larger population from which the sample was drawn.

Question 13

Steps to Hypothesis testing when testing. length of days suffered with headache?

Answer 13

1. Formulate a Null Hypothesis.
“In the population of patients studied with severe headache the proportion of days with headache is the same for analgesic and placebo”
2. Measure how far your sample data appear to
depart from the Null Hypothesis (test statistic). “Does our sample mean of 0.056 provide any
evidence that this is not true?”
3. Work out how unlikely a departure this big or bigger would be if the Null Hypothesis were true (p-value).

Question 14

What does the null hypothesis state?

Answer 14

The NH states that “No relationship exists between the variables and outcomes of the a study”

Question 15

What is the test statistic?

Answer 15

A statistical test compares what is observed (statistic) and what we would expect by chance alone (standard error (SE)).
Test statistic= 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛/𝐶h𝑎𝑛𝑐𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐/ 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

Question 16

How do you determine if the null hypothesis is rejected?

Answer 16

If observed statistic < SE
observed statistics was due to chance alone
✓ If observed Statistic > SE
chance explanation becomes less and less
plausible as this ratio (Test statistic) become greater and greater than 1

Question 17

What is the p value?

Answer 17

p-value is the probability of obtaining a test statistic result at least as extreme as the one that was actually observed, assuming that the null hypothesis is true
✓ Note if this is unlikely (p value very small) then we have evidence against the Null Hypothesis.
✓ The smaller the p-value, the stronger the evidence against the Null Hypothesis.
✓ It is common to describe p-values of less than 0.05 as providing evidence against the Null Hypothesis.

Question 18

What are the assumptions of One-Sample t-test

Answer 18

The sample is representative of the population.
✓ Measurements in the sample are all independent of one another.
✓ Measurements are Normally distributed in the population.
If the assumptions of a test are not met the p- value may not be correct.

Question 19

Why does 95% confidence interval work

Answer 19

Approximate 95% CI for population mean: sample mean + ( 1.96 x s.e. of sample mean)
* This works because the sampling distribution of the sample mean is approximately Normal.

Question 20

Why so we need to know the degrees of freedom?

Answer 20

It turns out we can get an exact 95% CI by using a different multiplier in place of 1.96:
sample mean + (t x s.e. of sample mean)
Obtain figure from Statistics table (column labelled “probability 0.05”).
To use table, need to know degrees of freedom (d.f.): here d.f. = sample size - 1.

Question 21

Example of hypothesis testing with
sample mean = 0.056; d.f. = 19; SE = 0.0138 multiplier (t)= 2.093

Answer 21

sample mean = 0.056; d.f. = 19; SE = 0.0138 multiplier (t)= 2.093
95% CI for population mean: 0.056±0.0138*2.093
=0.027 to 0.085
We can be 95% confident that in the population of patients studied with severe headache the difference in the proportion of days with headache between analgesic and placebo is somewhere between 0.027 and 0.085
NB – This appears to rule out the NH.

Question 22

When is the null hypothesis plausible?

Answer 22

Question 23

What is the difference between confidence intervals and p values?

Answer 23

CI - the set of all values for which the corresponding null hypothesis would not be rejected

Confidence intervals:
P values:
* Allows conclusions about whether null hypothesis is correct
* Does not allow conclusions about the size of the difference

Confidence intervals:
* Indicate the size of the difference and sample size
* Estimates the range of values calculated from the sample that is likely to contain the population mean
* May allow conclusions about statistical significance