Achieved result: 8.5
This is a comprehensive summary of the course “Statistics 2". Teaching material from the various lessons and related literature have been incorporated into it. In addition, various examples have been used throughout the summary as to provide clarification to the sometimes ...
,Tutorial 1: Normal(standard) distribution and calculating probability
(with samples)
Normal Distribution (introduction) is:
• Symmetrical
• Bell-shaped
• Uni-modal
• Parameters of the ND: determine shape
o 𝝁: mean of the population - is the expected value of y
▪ Medium of normal distribution
o y: the variable of interest - e.g. variation weight of an
apple
o 𝝈: standard deviation of population
E.g. weight of an apple normally distributed:
• Variable y is normally distributed with parameters 𝜇 and 𝜎 in the
population of apples.
• Examples of two different expressions:
o because the 𝝁 is lower with the elstar, the graph is depicted more to the lef-hand
side → determines the position!
o because the 𝝈 is lower with the elstar, the graph is more pointy. → determines the
width!
Examples of calculating probabilities with:
- What is the probability of P x<4 with a normal distribution with a mean of 5 (𝜇) and a
standard deviation of 2 (𝜎)?
- Answer:
𝑋− 𝜇 4− 5
1. Calculate the Z-value: 𝑍 = 𝜎 → 2 = -0.5
2. Check Z-table: -0.5 gives 0.3085
3. Since the P x< we don’t have to substract this value from a 1 (in case that the
question was posed with P x> then we still had to subtract the retrieved z-value
from 1.
,- Calculate the probability of a Standard Normal Distribution where the value is >1.5
- Answer:
1. Check what the mean (𝜇) and a standard deviation of (𝜎) are. → for a standard
normal distribution this is always 0 (𝜇) and 1 (𝜎).
𝑋− 𝜇 1.5− 0
2. Calculate the Z-value: 𝑍 = 𝜎 → 1 = 1.5
3. Check Z-table: 1.5 gives 0.9332
4. Since P x> then we still have to subtract the retrieved z-value from 1. → 1 –
0.9332 = 0.0668
- Calculate the probability of someone becoming 82 in a population where the average age is
76 (𝜇) with a standard deviation of 2.7 (𝜎).
- Answer:
𝑋− 𝜇 82− 76
1. Calculate the Z-value: 𝑍 = 𝜎 → 2.7 = 2.222222222222
2. Check Z-table: 2.2222222 gives 0.9868
3. Since P x> then we still have to subtract the retrieved z-value from 1. → 1 –
0.9868 = 0.0132
• Acrylamide: research question
o Research question: How much is the acrylamide content (μg/g) of baked potatoes
and what relation is there between acrylamide and other quality features?
o Important: What is our target group for which we need to answer this question?
▪ For example: all “home bakers” in the Netherlands
o Population = all households in the Netherlands that bake potatoes
o Unit: household (that bakes potatoes)
o Sample: selection of units from the population (for example: Simple Random Sample
= SRS)
o Variable: property of a unit from the sample
▪ Various variables are possible to be measured
> either:
> Qualitative:
o nominal (there is no natural order)
o ordinal
> Quantitative:
> discreet: can only take certain values (like
whole numbers)
> continuous (all possible outcomes would be
possible – within a range)
,▪ Visualization of quantitative variables:
Lots of observations also yields more classes and therefore more
nuance! The more classes the more the histogram becomes a curve →
the Probability Density Function
▪ Continuous random variable:
!! → 1 represents a 100% chance of it happening.
,• Normal distribution:
o Standard Normal distribution:
▪ Mean(𝜇) is 0
▪ Standard deviation (𝜎) is 1
,o Examples:
o Transformation to a standard Normal distribution: KEY
, o Calculation of probabilities:
!! – Just use 3 steps as depicted on page 2!
• Normal – Quantile Plot (Q-Q plot)
o Are observations normally distributed?
Key! →
o The example of the Acrylamide content: the population is definitely not normally
distributed as can be seen below! → Dots are not on the line
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