Confidence Intervals & Z-Scores Statistics is the grammar of - - PowerPoint PPT Presentation

Confidence Intervals & Z-Scores

“Statistics is the grammar of science.”

Karl Pearson (Mathematician)

“How confident are we in our statistic?” Standard error – standard deviation of a statistic Standard error of the mean - reflects the overall distribution of the means you would get from repeatedly resampling

n = sample size s = sample standard deviation

Standard error

Small values = the more representative the sample will be of the overall population Large values = the less likely the sample adequately represents the overall population

𝑇𝐹𝑦 = 𝑡 𝑜

The t-distribution (a.k.a The Student t-distribution)

t-distribution

(sampling distribution)

Normal distribution

William Sealy Gosset (1876-1937)

𝑒𝑔 = 𝑜 − 1

• Has fatter tails then the normal distribution
• Degrees of freedom:
• As sample size increases – it approaches the normal distribution
• Properties:

 Bell-shaped  mean=median=mode=0  Variance > 1

Confidence Intervals

𝑇𝐹𝑦 = 𝑡 𝑜

Based on this curve:

• 68.27% confident that the true mean is within 1 𝑇𝐹𝑦of 𝑦
• 95.45% confident that the true mean is within 2 𝑇𝐹𝑦 of 𝑦
• 99.73% confident that the true mean is within 3 𝑇𝐹𝑦 of 𝑦

For confidence intervals:

• 95% confident that the true mean is within 1.96 𝑇𝐹𝑦 of 𝑦

𝐷𝐽95 = 𝑦 ± 𝑇𝐹𝑦 ∗ 1.96

Standard deviation vs Standard error

𝑇𝐹𝑦 = 𝑡 𝑜 68% of raw data falls within here 68% confident that the true mean is in here

Standard Error

(inferential statistics)

“trying to draw conclusions”

Standard Deviation

(descriptive statistics)

“trying to describe data”

“What should I display as part of my results?” “What message do I want to give?” 𝑡2 = 𝑦𝑗 − 𝑦 2

𝑜 𝑗=1

𝑜 − 1 s = 𝑡2 𝐷𝐽68 = 𝑦 ± 𝑇𝐹𝑦

More Vocabulary

• Percentile – the value below which a given percentage of observations within a

group fall

• Quartile – (1st, 2nd, 3rd, 4th) points that divide the data set into 4 equal groups, each

group comprising a quarter of the data

• Alpha level – predetermined probability where we make some sort of decision
• P-value – (percentiles) the probability the observed value or larger is due to random

chance

• Critical t-value – the t-value that corresponds to the 𝛽 − 𝑚𝑓𝑤𝑓𝑚
• Actual t-value – the t-value that corresponds to and raw data value being tested

with the 𝛽−𝑚𝑓𝑤𝑓𝑚 (signal-to-noise ratio)

• Signal – the difference between the test and mean values
• Noise – measure of the distribution of the data

T-value

(standard error)

P-value

(percentiles, probabilities)

(t-𝑤𝑏𝑚𝑣𝑓 ∗ 𝑇𝐹𝑦) + 𝑦 Original units 𝑟𝑢(𝛽, 𝑒𝑔) (𝑤𝑏𝑚𝑣𝑓 − 𝑦 )/𝑇𝐹𝑦 p𝑢(t−𝑤𝑏𝑚𝑣𝑓, 𝑒𝑔)

• 1

1 2 3

• 2
• 3

0.001 0.999 0.50

How to convert between scales

𝑦

Raw data value Critical t-value -level

Z-score

(standard deviation)

(z-𝑤𝑏𝑚𝑣𝑓 ∗ 𝑡) + 𝑦 Original units (𝑤𝑏𝑚𝑣𝑓 − 𝑦 )/s

• 1

1 2 3

• 2
• 3

Z-scores

𝑦

Raw data value Z -value

• Similar to the t-distribution for large sample sizes (N≥30)
• Need to know the population standard deviation is known
• Why we use the t-distribution for statistics
• But we can use it to define where a given value falls within a distribution of

values in terms of standard deviations away from the mean