Question

In 10-mph crash tests, 25% of a certain type of automobile sustain no visible damage. A modified bumper design has been proposed in an effort to increase this percentage. Let denote the proportion of all cars with this new bumper that sustain no visible damage in 10-mph crash test. The hypothesis to be tested is The test will be based on an experiment involving independent crashes of car prototypes with the new bumper. Let denote the number of crashes resulting in no visible damage, and consider the test procedure that rejects

(a) Find the probability of type I error.

Answer 1

The probability of Type I error is = 0.10185.

Explanation:

Solution:-

The type I - error is defined as the probability of rejecting Null hypothesis defined by Alternate hypothesis:

Ha : X ≥ 8

Where,

X : Denote the number of cars crash with no visible damage

The random variate "X" is defined by binomial distribution:

X ~ B ( n = 20 , p = 0.25 )

- The probability of Type I error:

P (Type I error ) = P ( Reject Null hypothesis )

= P ( X ≥ 8 )

- The probability mass function of binomial random variate "X" is given:

`P ( X = x ) = nCr (p)^r * (1-p)^(^n^-^r^)\\P ( X \geq 8 ) = 1 - P ( X < 8 )\\\\P ( X \geq 8 ) = 1 - [ P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 ) + P ( X = 5 ) + P ( X = 6 ) + P ( X = 7 ) ]`
`P ( X \geq 8 ) = 1 - [ (0.75)^2^0 + 20(0.25)*(0.75)^1^9 + 20C2(0.25)^2*(0.75)^1^8 +\\\\ 20C3(0.25)^3*(0.75)^1^7 + 20C4(0.25)^4*(0.75)^1^6 + 20C5(0.25)^5*(0.75)^1^5\\\\ + 20C6(0.25)^6*(0.75)^1^4 + 20C7(0.25)^7*(0.75)^1^3 ] \\\\\\P ( X \geq 8 ) = 1 - [ 0.00317 + 0.02114 + 0.06694 + 0.13389 + 0.18968 + 0.20233\\\\+ 0.16860 + 0.11240]\\\\P ( X \geq 8 ) = 1 - 0.89815 = 0.10185`

Answer: The probability of Type I error is = 0.10185.

Answer 2

(a) Find the probability of type I error. = 0.1018

Explanation:

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