Sampling Variation and Confidence Intervals Flashcards
What does the central limit theorem state?
The central limit theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size increases.
What is the statistic that tells us how representative our sample is likely to be compared to the overall population?
This statistic is called the standard error of the means, and it is simply the standard deviation of the sampling distribution.
What equation is used to calculate the standard error of the mean (SE)?
SE = s / (square root of N)
Where s is the sample standard deviation, and N is the sample size.
What does a large value for the standard error of the mean (SE) imply?
A large value will tell us that the sample means can be quite different to each other, and therefore a particular sample may not be particularly representative of the population.
What does a small value for the standard error of the mean (SE) imply?
Small values for the standard error of the mean tell us that the sample means will be fairly similar, and therefore our particular sample is likely to be a fair selection of the population.
What measure could you use to define the limits between which we would expect most of the sample means to fall?
As well as calculating the standard error of the mean, we could also use confidence intervals to define the limits between which we would expect most of the sample means to fall. Most studies will use 95% confidence intervals, although you may sometimes see 99% confidence intervals reported.
If the mean of your sample represents the data well, then will the confidence interval of that mean be small or large?
If the mean of your sample represents the data well, then the confidence interval of that mean should be small, much like the standard error of the mean.
What does a wide confidence interval for a mean imply?
If the sample mean is a bad representation of them population, then the confidence interval will be wider, indicating that a different sample might produce a mean quite different from that of our original sample.
How can the confidence intervals of a sample be calculated?
The confidence intervals can be calculated once the standard error is known.
The 95% confidence interval is simply the mean plus/minus 1.96 multiplied by the standard error.
What is the equation for calculating a 95% confidence interval?
95% confidence interval = mean +/- 1.96 x SE
Where does the figure 1.96 come from that is used in the confidence interval equation?
1.96 comes from a normal distribution. It actually represents the number of standard deviations away from the mean which encompasses 95% of the population.
What is a t-distribution?
When the sample size is small, we should allow for the fact that even our estimate of the standard deviation (SD) will have some error in it. We do this by using a slight adaptation of the normal distribution, called a ‘t-distribution’ and we use the equivalent 95% cut off from this distribution instead. However, generally we get confidence intervals from SPSS and this does all the work for us.
If we have a sample with a mean of 300 and the 95% confidence interval goes from 293 to 307 how should we write this?
mean (95% CI) = 300 (293, 307)
If you repeated an experiment 100 times and calculate the 95% confidence interval each time, how many of these samples would we expect to contain the true population mean?
If you repeated the entire experiment 100 times, and calculated the 95% confidence interval each time, we would expect 95 of these 100 samples to contain the true population mean.
We can only calculate confidence intervals for the mean. True or false?
False. Whenever we take a sample we are estimating a population parameter, and our estimate will have a sampling error. We can calculate a confidence interval for all different types of estimates, and it means exactly the same thing as for sample mean.
