Lecture 6 - Confidence interval and hypothesis testing

Question 1

What is a confidence interval?

Answer 1

• Statistics means never having to say you’re 100% certain
• Statements must be qualified as any data is imperfect
• In any research, all we have is the sample data
• But we want to know more
– We know what happened with the sample, but what about the wider population of interest?
• We use information from the sample statistic to make inference about the population parameter

Question 2

Parameters vs Statistics

Answer 2

• Parameters are
– Numerical characteristics of a population
– Constant (fixed) at any one moment – Usually unknown
• Statistics
– Numerical summary of a sample
– Calculated from sample data (not constant) – Used to estimate a parameter

Question 3

What is a confidence interval?

Answer 3

• Using sample data to make inference to the wider population has limitations
– After all you only have one sample and not all samples will be the same
• So, all you can do is to estimate
• The statistic calculated from the sample is a point estimate of the corresponding population parameter
– The sample average is a point estimate of the true population mean
– The sample proportion is a point estimate of the population proportion
• Interval Estimate
– Range of values which is likely to contain the population
parameter
• Confidence intervals are constructed to provide information about the precision of the estimate
– A confidence interval is an interval within which we can estimate, with some confidence, that the true population parameter lies
• The sample statistic is the sample mean ( x ) which is a point estimate of the true population mean (μ)
– Because the mean can vary from sample to sample, this estimate from one sample does not necessarily equal the true mean
• Confidence Intervals provide a measure of the precision of the estimate of the mean from one sample

Question 4

What is the confidence interval based on?

Answer 4

• Point estimate
• Desired level of confidence
– Common CIs are 90%, 95% and 99%
• Sampling variability or the standard error of the point estimate

Question 5

Interpretation of confidence intervals

Answer 5

• Correct interpretation
– If we were to select 100 samples from the population and use these samples to calculate 100 different confidence intervals for μ, approximately 95 of the intervals (95%) would include the true population mean
• Incorrect interpretation
– The probability that the true mean (μ) is in the interval = 0.95

Question 6

Confidence intervals of the mean using Z-scores

Answer 6

• Z-scores are appropriate confidence coefficients for a confidence interval of the mean when the population standard deviation (σ) is KNOWN
• However, most of the time when the population mean is being estimated from sample data the population variance is unknown and must also be estimated from sample data
• The sample standard deviation (s) provides an estimate of the population standard deviation (σ)

Question 7

Confidence intervals when the population variance is unknown

Answer 7

• The sampling distribution of the sample mean is a t-distribution with n-1 degrees of freedom instead of a normal distribution
• Instead of using the standard normal distribution to find confidence coefficients, the t-distribution with n-1 df is used to find the confidence coefficients which are called t- coefficients or t-critical values

Question 8

Hypothesis testing

Answer 8

• Method of making decisions using data
• Two Hypotheses:
– Null: Hypothesis of no difference
– Alternative: researcher’s hypothesis
• Analysis of the data using statistical tools
• Use of the significance level (α) for the decision rule
• Decision: REJECT or FAIL to reject (No sufficient evidence to reject) the Null hypothesis

Question 9

Steps in hypothesis testing

Answer 9

• State the research question
• Specify the null and alternative hypothesis
– In hypothesis testing, we assumed null hypothesis is true and try to find evidence to support the alternative
– Difficult to be absolutely confident something works
• If I cant prove, it doesn’t work, the logical alternative is, it must work
– E.g.: guilty vs not guilty
• Decide on the significance level
– a statistical assessment of whether observations reflect a pattern rather than mere chance
– Think about type I and II error
• Calculate test statistic – Z, t or other statistic
• Decide whether to reject or fail to reject the null hypothesis (use of p value)
• State the conclusion