Question

1. Kaizen Consulting Group (KCG) are testing a new design for deep fryers to be installed at Beaver Stadium. Currently, the fryers have the capacity of frying an average of 300 orders of chicken tenders per cook. This capacity resulted in an average wait time of 18 minutes for a customer to get an order during the 2018 football season. Kaizen will suggest replacing the current fryers only if sample data strongly suggests the fryers are not meeting their current (stated) capacity.
   1. Which hypotheses should be tested to determine if the current fryers are not meeting capacity?
   2. Suppose the capacity of the fryers is normally distributed with a σ=20. Between classes and during the weekends, KCG collected a random sample of 25 observations, and calculated xbar for the sample capacity mean. What is the probability associated with committing a Type I error if the critical region used would be xbar <= 290? Explain, in words, what committing a Type I error means for this situation. Depict the Type I error probability graphically.
   3. What is the probability of committing a Type II error if the actual (average) capacity of the fryer is 285 orders of chicken tenders per cook and the critical region from part (b) is used? Explain, in words, what committing a Type II error means for this situation. Depict the two error probabilities graphically.

Answer 1

Solution

Back-up Theory

Type I Error is the error of rejecting a null hypothesis when it is true…………...........................................………………. (1)

Type II Error is the error of accepting a null hypothesis when it is not true, i.e., Alternative is true (2)

α = P(Type I Error) = probability of rejecting a null hypothesis when it is true ……..........................................…………. (1a)

β = P(Type II Error) = probability of accepting a null hypothesis when it is not true, i.e., Alternative is true. ……………. (2a)

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then, Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution …………………..............................................................……..(3)

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .…....................................................................……(3a)

X bar ~ N(µ, σ²/n),…………………………………………...............................................................………………….…….(3b),

where X bar is average of a sample of size n from population of X.

So, P(X bar ≤ or ≥ t) = P[Z ≤ or ≥ {(√n)(t - µ)/σ }] …….............................................................……………………………(3c)

Probability values for the Normal Variable, X ~ N(µ, σ²), can be found using

Excel Function: Statistical, NORMDIST, which gives P(X ≤ t) ……………...........................................................….…..(3d)

Now to work out the solution,

Let X = Number of tender chicken fried per cook by the existing fryers. Let mean and standard deviation of X be µ and σ.

Part (a)

Hypotheses:

Null H₀: µ = µ₀ = 300   Vs Alternative H_A: µ < 300   Answer 1

[H₀ => existing fryers are meeting the specification and H₁ => existing fryers are not meeting the specification]

Part (b)

Given X ~ N(µ, σ²) with σ = 20, n = 25 and Critical region (CR) as Xbar ≤ 290.

Vide (1a),

P(Type I Error) =

= P(Xbar ≤ 290/µ = 300, σ = 20, n = 25)

= P[Z ≤ {(√25)(290 - 300)/20 }] [vide (3c)]

= P(Z ≤ - 2.5)

= 0.0062 [vide (3d)] Answer 2

Interpretation

In given scenario, there is just 0.62% chance of concluding that the existing fryers are not meeting the specification, when in reality existing fryers can meet the specification. Answer 3

Part (c)

Vide (2a),

P(Type II Error) =

= P(Xbar > 290/µ = 285, σ = 20, n = 25, note under H_A: µ = 285 given)

= P[Z > {(√25)(290 - 285)/20 }] [vide (3c)]

= P(Z > 1.25)

= 0.1056 [vide (3d)] Answer 4

Interpretation

In given scenario, there is 10.56% risk of concluding that the existing fryers are meeting the specification, when in reality existing fryers cannot meet the specification. Answer 5

DONE