Inferential statistics, Cheat Sheet of Building and Prefabrication

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INFERENTIAL STATISTICS

LECTURE 1

NORMAL DISTRIBUTION

• This is the probability distribution of random continuous variables
• We observed that probability is a form of frequency distribution
• Relative frequency distribution totals to 1
• Probability distributions also total to 1
• The edges of a frequency histogram can smoothen out into a curve (or a smooth frequency polygon)
• The area under the curve can be assumed as equal to 1
• This leads us to the probability density curve
• The mathematical function f(x) whose graph produces this curve is called the probability density function

• Data can be distributed or spread out in many ways

1. Skewed-Left or long-tail left
2. Skewed-Right or long-tail right
3. Uniform (flat)
4. Peaked
5. Jumbled
6. Bell shaped (normal distribution)

• Normal distribution is a case where the data tends to be around a central value with no bias left or right

Mean and Standard Deviations

• Interval Probabilities:
• One sd each side of the mean
• P[μ - σ < X < μ + σ ] = 0.
• Two sd each side of the mean
• P[μ - 2 σ < X < μ + 2 σ ] = 0. 954
• Three sd each side of the mean
• P[μ - 3 σ < X < μ + 3 σ ] = 0. 997

Standard deviations in a normal

distribution

• 68.3% of values are within fall within 1 standard deviation of the mean or probability of 0.
• 95.4% of values are within fall within 2 standard deviations of the mean or probability of 0.
• 99.7% of values are within fall within 3 standard deviations of the mean or probability of 0.

• Worked Example :
• In a primary school 95 % of students are between 1.1m and 1.7m tall. Assuming this data is normally distributed can you calculate the mean and standard deviation?
• The mean is halfway between 1.1m and 1.7m:
• Mean = 1.1m + 1.7m 2 = 1.4m
• 95 % is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:
• 1 standard deviation =
  1. 7 𝑚− 1. 1 𝑚 4 = 0. 6 𝑚 4 = 0.15m

• Formula for z-score or standardizing is:
• z = x − μ 𝜎
• Where:
• z is the "z-score" (Standard Score)
• x is the value to be standardized
• μ is the mean
• σ is the standard deviation
• The value of Z indicate that the observation being compare with the mean is smaller than the mean (for negative values) or greater than the mean for negative values

Standardize the Normal Curve

• Standardize the normal curve from the earlier example using the formula:
• Where
• 𝜎 = 0. z = x − μ 𝜎

Standard Normal Tables

• The standard normal curve can lead us to the standard normal tables
• The area under the curve could be computed using integral calculus
• However for purposes of simplifying statistical calculations standard normal tables have been derived to compute the area under the normal curve
• The areas so calculated form proportions of the entire distribution or the entire populations from which inferences may be made
• The proportions can be used to estimate the probabilities that an observation in the normal population will be found within the slice being considered

• The standard normal tables are made in such a way that if the number of standard deviations from the mean to any individual observation can be calculated then the area under the normal curve between the mean and the observation can be looked up in the standard tables
• The tables therefore mark an automatic transformation of any actual normal curve into standard normal curve.

Z Score Values

• Z score can take negative or positive values so there is a table for negative values and another table for positive values.
• Corresponding values which are less than the mean are marked with a negative score in the z-table and represent the area under the bell curve to the left of z.
• Use the positive Z score table below to find values on the right of the mean as can be seen in the graph alongside. Corresponding values which are greater than the mean are marked with a positive score in the z-table and respresent the area under the bell curve to the left of z.