Topic 4: Normal Model

Question 1

What is the normal curve?

Answer 1

It is a probability distribution symmetric about the mean, showing that data around the mean are more frequent in occurrence than data far from the mean

It is also recognised as a ‘bell curve’

Discovered by De Moivre

It is the probability density function (pdf) - f(x). The PDF is a special function describing the chance associated with a continuous variable X, over all values of X

Question 2

What are the two types of normal curves?

Answer 2

The standard normal curve (z)
The general normal curve (x)

Question 3

What is the standard normal curve?

Answer 3

It has a mean of 0 and SD of 1

Question 4

What is the general normal curve?

Answer 4

It can have any mean and SD

Question 5

Which scenario allows us to use the normal curve as an approximation to the area under the histogram?

Answer 5

When the normal curves seems to fit the histogram

Question 6

What is the normal curve notation?

Answer 6

X ~ N(μ, σ^2)

Question 7

What are the 2 special properties of a normal curve?

Answer 7

1) All normal curves satisfy the ‘68%-95%-99.7%’ rule

2) Any general normal can be rescaled into the standard normal through the use of standard units (z score)

Question 8

What does the z score measure

Answer 8

How many SDs a point is above or below the mean

Question 9

What is the 68%-95%-99.7% rule?

Answer 9

Area under 1sd out from the mean in both directions is 68% of total values

Area under 2sd out from the mean in both directions is 95% of total values

Area under 3sd out from the mean in both directions is 99.7% of total values

Question 10

How is standard unit (z score) calculated?

Answer 10

(data point - mean) / SD

Question 11

What are the limitations of using statistical models?

Answer 11

All models are approximations, and thus they are all wrong. However, only some models are useful, so we have to ask ‘which model is best for this particular application’?

Question 12

How do we know when to use the normal curve?

Answer 12

Does the histogram look normal?

Do the proportions look right?

Does the quantile - quantile (QQ) plot look like a straight line?

Shapiro Test

Question 13

What does it mean by ‘does the histogram look normal?’

Answer 13

Check whether the histogram looks bell shaped, and if it has any outliers or long tails –> if yes we can use the normal approximation/curve / distribution

Question 14

What does it mean by ‘do the proportions look right?’

Answer 14

Involves checking does 1sd above and below the mean give us ~68% of results? and continuing on for other sds –> if yes we can use the normal approximation/curve / distribution

Question 15

What does it mean by ‘does the quantile-quantile (QQ) plot look like a straight line?’

Answer 15

A Q–Q plot is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against each other. A point on the plot corresponds to one of the quantiles of the second distribution plotted against the same quantile of the first distribution.

If it looks like a straight diagonal line, then we can use the normal curve/approximation / distribution

Question 16

What does it mean by ‘Shapiro test?’ What value on the shapiro test is ideal?

Answer 16

The Shapiro test essentially just tests for normality / does a sample fit a normal distribution?

A p value of below 0.05 means that the normal doesn’t fit, if it is above 0.05, we can have assume a normal distribution / approximation / curve

Question 17

What is measurement error

Answer 17

Measurement Error (also called Observational Error) is the difference between a measured quantity and its true value.

Question 18

What is the equation to account for measurement error?

Answer 18

Individual measurement = exact value + chance error + bias

An individual measurement often differs from the exact value

Question 19

What is chance error?

Answer 19

No matter how carefully any measurement is made, it could have turned out differently just slightly. This is due to chance error

“Chance error” typically refers to the variability or random fluctuations that occur in the measurement of a quantity due to random factors or chance events. It is also known as random error. Chance errors are inherent in any measurement process and can result from a variety of unpredictable factors, such as instrument limitations, environmental conditions, or human error.

Unlike systematic errors, which are consistent and repeatable, chance errors are unpredictable and can affect measurements differently each time they are taken.

Question 20

What is the best way to estimate chance error?

Answer 20

Replicate the measurement under the same conditions, and calculate the standard deviations

Question 21

What is an outlier?

Answer 21

It is an observation that lies at an abnormal distance from other values in a random sample from a population

It is statistically defined as a value more than 3ssd away from the mean, assuming normality

Question 22

What is a measurement bias? WHat are the features?

Answer 22

It is a confounding variable which typically leads to a systematic error. A measurement is considered biased if it systematically overstates or understates the true value of a measurement

Bias can be deliberate or accidental

Results in a constant amount being added or subtracted from each measurement

Question 23

What is an example of measurement bias?

Answer 23

If a scale isn’t properly calibrated. In this case the scale is producing the bias

Question 24

Can bias be estimated by replicating the measurement?

Answer 24

No, it can’t because a systematic error will still remain

Question 25

Does mean or sd change with an increase or decrease in price?

Answer 25

Mean changes according to the decrease or increase, meanwhile there is no sd change