Section 4 (start of section 2 - p59) Flashcards
What is the approach used in statistical inference when the purpose is to obtain information about the population parameters, such as the mean and standard deviation?
Estimation
What is the approach used in statistical inference when the purpose is to make comparisons with some hypothesised value?
Hypothesis testing
What are the 2 types of estimate?
Point
Interval
How are point estimates of population parameters derived?
From the corresponding sample parameters
When is a sample parameter said to be an unbiased estimator of the population parameter?
If the average of all possible sample parameters is equal to the population value
How is an estimator of a population parameter represented in symbol form?
By the symbol for the parameter with a hat above it
How is the uncertainty associated with a point estimation expressed?
By confidence intervals
What is a confidence interval?
A range which we would expect, with a given level of confidence, to include the population parameter
What is the usual level of confidence used?
Other possible values?
95%
Can also use 99% and 99.9%
What happens to the width of the confidence interval as the level of confidence increases?
Width also increases
What is the name for the upper and lower values for the confidence interval?
Confidence limits
How are the confidence limits obtained?
By adding/ subtracting a values to/ from the value of the point estimate
Describe how the confidence interval changes for more or less variable data?
The less variable the data the narrower the confidence interval
What does precision refer to?
The variability of an estimate, not its accuracy
What does hypothesis testing use instead of an interval to express information about the population parameter?
A numerical value called the test statistic
What is the name of the statement based on the test statistic that is used to determine whether a claim about a population parameter, made in the null hypothesis, is accepted or rejected?
Decision rule
What is the symbol for the null hypothesis?
H0 (H nought)
What idea does the null hypothesis usually express?
There is no effect
What is tested against the null hypothesis?
The alternative
Symbol for the alternative hypothesis?
H1
What is the standard error?
Standard deviation of the mean
Is the mean of a sample from a non-normal population normally distributed?
Approximately - the larger the sample, the better the approximation
What is the fact that the mean of a sample from n independent items from a non-normal population can be described as approximately normal, with the approximation being better the larger the sample is?
Central limit theorem
What is the main concept of the central limit theorem?
No matter what shape of sample you have, if the variables are independent and random, the average of the means will be normally distributed, if the sample size is large enough
How large does the sample size need to be for the central limit theorem?
about 30 samples if the population distribution is roughly bell-shaped
At least 40 if the original population is distinctly not normal
What factors are required for the mean of a sample from a normal population to be normally distributed?
Known population standard deviation
Observations in the sample are independent
What factors are required for the mean of a sample from a not normally distributed population to be normally distributed?
Large enough sample
Independent items
What is another name for plausibly?
Approximately
What would we expect to happen to the standard error and the confidence interval for the sample mean as the number in the sample increases?
Standard error decreases
Confidence interval becomes narrower
If the population standard deviation is unknown, the mean of a sample of n items from a normal population with mean mu has what distribution? Describe properties of this value?
T distribution with mean mu and standard error s/ square root of n
Describe the normal and t distributions when n is large?
Almost identical
What happens to the distribution of the sample proportion as the number in the sample increases?
Tends towards normality
When is the distribution of the sample proportion plausibly normal?
If both np and n(1-p) are greater than 5
For estimating proportions from sample parameters, what does the mean equal?
Population proportion p
For estimating proportions from sample parameters, what does the standard error equal?
Square root of p(1-p)/ n
What is the aim of a hypothesis test?
To asses the validity of a claim about a population parameter
When must the hypothesis to be tested be defined?
Before data is collected
Should you ever use one-sided hypothesis tests?
No, they are rarely justifiable
What is the area called when a random sample lie in the 5% chance area?
Critical region
What happens if a value lies in the critical region when doing a hypothesis test?
Casts doubt on the validity of the null hypothesis, which would then be rejected in favour of the alternative
What is it called when you reject a true null hypothesis?
Type I error
What is it called when you accept a false null hypothesis?
Type II error
What happens to type 2 errors if you reduce type 1 errors?
Increase type 2 errors (unless you increase the sample size)
What is the level of significance of the test?
The probability of making a type I error
What level of risk is considered to be acceptable for failing to detect an effect?
20%
What is the complement of the significance level?
The confidence level
Why don’t we always use a 1% significance level instead of a 5% significance level?
This would increase the chance of making a type II error (if we wrongly reject a true null hypothesis, we wrongly accept a false alternative - if we decrease the chance of one happening, we increase the chance of the other happening - need to strike a balance)
What level do we normally set the probability of making a type I error?
5%
What level do we normally set the probability of making a type II error?
20%
What is a test statistic?
A measure of the difference between what is expected if the null hypothesis were true and what is obserrved
What is the general formula for a test statistic?
Observed value of parameter - expected value of parameter/ standard error of parameter
What is the test statistic based on the normal distribution for means when the population standard deviation is know?
Z
What is the test statistic used for means when the population is normally distributed but the population standard deviation is not know?
Student t test for mean
Test statistic used for variance?
Fisher’s f test for variance
Test statistic used for proportions?
Chi squared test for proportion (x^2)
What is the decision rule?
A statement of the conditions (values of the test statistic) for which the null hypothesis will be rejected
Decision rule number for 5% level of significance?
1.96
Decision rule number for 1% level of significance?
2.58
What are the 8 steps for testing a hypothesis?
- Identify the distribution of the data
- Construct the null and alternative hypothesis
- Establish the significance level
- Identify the test statistic
- Formulate the decision rule
- Carry out the study
- Conduct the test
- Make the decision and interpret the result
