Wk 8: All About Means

Question 1

4x participants will ______ (equal/double/triple/half) the accuracy of the results because the variance of the difference of the means between the two groups will halve.

Answer 1

double

Question 2

What are 3 steps to measure the quality of prediction value?

Answer 2

1. Measure prediction errors - the difference between your prediction value and each existing data point.
   
   • x = prediction value (can be any number)
   • x̅ = prediction value that minimises the sum of squared prediction erros = sample mean
2. Square each prediction errors (so they will be positive numbers for differentiation later)
3. Add them all together: When the sum of squared prediction error is 0, it means the prediction value is most accurate.

Question 3

What is the equation of sample mean?

Answer 3

• x̅ = sample mean
• x1, x2 etc. = value of each data point
• n = number of data

Question 4

What is the equation of sample standard deviation?

Answer 4

Question 5

The sample mean is an ______ of the population mean

Answer 5

unbiased estimator

Question 6

What are the 3 features of smaple mean as an unbiased estimator of the population mean?

Answer 6

1. The variability (spread) of this estimate becomes smaller as the sample size (n) increases (proportional to 1 = √n).
2. This implies that the sample mean is a more precise estimator of the population mean for larger samples.
3. The shape depends on sample size. The larger the sample size, the more symmetrical and concentrated the shape.

Question 7

What are 2 features of robustness?

Answer 7

1. Outliers can affect the sample mean so always visualise data first (boxplots) before analysis.
2. Median is a more robust measure of centrality and skewed distributions

Question 8

Outliers can affect the _____ so always visualise data first (boxplots) before analysis.

Answer 8

sample mean

Question 9

_______ is a more robust measure of centrality and skewed distributions

Answer 9

Median

Question 10

What are 3 features of outliers?

Answer 10

1. Note the outliers and dig into it
2. If they represent the population as an atypical member, then explore why they are a outlier
3. If they do not represent the population, then exclude them from analysis

Question 11

What are 2 features of smaple variance?

Answer 11

1. Range
2. Interquartile range (suitable for more data)

Question 12

Sum of squared prediction errors is ______. What does that mean?

Answer 12

sample variance

• But if there is more data, then the sum of squared prediction errors will increase, therefore we need normalisation

Question 13

What is normalisation?

Answer 13

Divide the sum of squared prediction errors by n-1.

Question 14

What are 3 features of normalisation?

Answer 14

1. As the standard deviation from sample mean is always 0, we could deduce the value of the last one, so there is n-1 pieces of independent information.
2. Degrees of freedom: The number of independent pieces of information.
3. Need to take square root afterwards so the units are not squared.

Question 15

What is parameter?

Answer 15

a numerical characteristic of a population (e.g. Height, weight).

Question 16

What is a statistic?

Answer 16

We use statistics to estimate unknown population parameters.

Question 17

________ describes the shape of a population distribution

Answer 17

Normal (Gaussian) distribution

Question 18

What are 3 features of Normal (Gaussian) distribution?

Answer 18

1. Describe the distribution of observations
2. Describe the distribution of statistics (e.g. Sample mean, sample proportion
3. Normal distribution is determined by mean (location) and standard deviation (spread).

Question 19

What does Normal quantile-quantile plot compare to?

Answer 19

the quantiles from our data against the quantiles of the theoretical normal distribution.

• If out data is normally distributed, it should lie on the black line on normal quantile-quantile plot.

Question 20

What is the 68-95-99.7 Rule?

Answer 20

1. If out data is normally distributed, its histogram should follow a bell curve.
2. Within 1 standard deviation of the mean are 68% of the values (area under the bell curve)
3. Within 2 standard deviations of the mean are 95% of the values
4. Within 3 standard deviations of the mean are 99.7% of the values

Question 21

What are 2 features of Central Limit Theorem?

Answer 21

1. If the population is normally distributed, then the sample mean is normal for any sample size.
2. If the population is not normally distributed, then the sample mean is still approximately normal, and gets more normal as the sample size increases

Question 22

What are 5 features of confidence intervals?

Answer 22

1. Confidence interval is an interval estimate that might contain the true value of an unknown population parameter.
2. Confidence level (usually 95%) is the probability that any data from the population will land within the confidence interval.
3. Margin of error of the interval is one side of the confidence interval.
4. Increased sample size will narrow the confidence interval.
5. A higher confidence interval is wider on the bell curve.
   
   • 99% confidence interval is ~33% wider than a 95% confidence interval.
   • 100% confidence interval is infinitely wider than a 95% confidence interval.

Question 23

What is a confidence interval?

Answer 23

an interval estimate that might contain the true value of an unknown population parameter.

Question 24

What is a confidence level?

Answer 24

(usually 95%) is the probability that any data from the population will land within the confidence interval.

Question 25

What is a margin of error of the interval?

Answer 25

one side of the confidence interval.

Question 26

Increased sample size will _____ (narrow/widen) the confidence interval.

Answer 26

narrow

Question 27

A higher confidence interval is ____ (wider/narrower) on the bell curve. What is the confidence interval?

Answer 27

wider

• 99% confidence interval is ~33% wider than a 95% confidence interval.
• 100% confidence interval is infinitely wider than a 95% confidence interval.