15 - SE of the Mean, Type 1, Type 2 errors

Question 1

What condition makes sample means from a distribution normally distributed?

Answer 1

n > 10 |K₄|

n = sample size

|K₄| = kurtosis

• from the CLT, we know that under certain conditions (iid sampling) sample means from a distribution will be normally distributed
  
  • use number of observations per sample, not the entire data set, to test this condition

Question 2

Standard Error of the Mean

Answer 2

SE(Xbar)

• SD(Xbar) = SE(Xbar)
• we get this from the CLT, remember that
  
  • Xbar₁₀₀ ~ N(µ, σ²/n)
    
    • µ = pop mean
    • σ²/n = pop variance
  • so, the SD of Xbar is σ/sqr(n)

Question 3

sample means are less variable than raw data by a factor of _____

Answer 3

1/sqr(n)

Question 4

UCL & LCL

Answer 4

UCL = µ + L

LCL = µ - L

*process in control if all x-bars are in this range*

Question 5

Type 1 Error

Answer 5

if falling outside the control limits = positive,

Type 1 = a false positive

Question 6

Choosing L

Answer 6

we want L such that when the process is in control,

P(µ-L <= Xbar <= µ+L) = High probability

→so there’s low prob of falling outside the limits and thus a low prob of Type 1 error

*if we want a 100a % chance of Type 1 error, then:

L = z_a/2SE(Xbar),

where z_a is the 100(1-a) percentile of the standard normal distribution*

Question 7

Ex. find control limits with a 1% chance of producing a Type 1 Error

(if µ = 8, σ = 1.2, and 100 observations a day)

Answer 7

L = z_a/2SE(Xbar)

z_a/2 = z_0.01/2 = z_.005

(z_.005 is the 99.5 percentile of the standard normal)

from z-tables, z_0.005 = 2.5758

SE(Xbar) = 1.2/sqr(100) = 0.12

L = 2.5758 * 0.12 = 0.309

LCL = 8 - 0.309 = 7.691, UCL = 8 + 0.309 = 8.309

Question 8

Xbar chart

Answer 8

when we track sample means

Question 9

Type 2 errors

Answer 9

same as a false negative

claiming that the process is in control, when it isn’t

Question 10

Type 1 and 2 in context of a production line

Answer 10

Type 1 = shut down line when you shouldn’t have

Type 2 = keep line going when you should’ve shut it down

Question 11

Tradeoff between Errors

Answer 11

For a fixed sample size, if you want a smaller Type 1 error then you must accept a larger Type 2 error

Question 12

S-chart

Answer 12

tracks sample SD’s over time