4, A certain type of automobile is known to sustain no visible damage 25% of the

Question

4, A certain type of automobile is known to sustain no visible damage 25% of the time in 10mph crash tests. A modified bumper design has been proposed in an effort to increase this percent age. Let p denote the proportion of all 10mph crashes with new bumper that result in no visible damage. The test will be based on an experiment involvingn-20 independent crashes with prototypes and the rejection region R 8,9,10 20 (a) State the statistical hypotheses for the test. (b) Let X denote the number of crashes with no visible damage. What is the distribution of X under Ho? Explain your answer. (c) what is the probability of making a type 1 error, i.e., P{reject HolHo true)? (d) What is the probability of making a type II error, ie., Pfdon't reject HolHo false) when 0.3? (17 marks)

Explanation / Answer

Solution

Given X = Number of crashes with no visible damage in 10 mph crash test with new bumper.

Also given

sample size = 20

known proportion of crashes with no visible damage in 10 mph crash tests currently = 0.25

Back-up Theory

X ~ B(n, p), where n = sample size and p = proportion of crashes with no visible damage in 10 mph crash tests with new bumper.

Part (a)

Hypotheses:

Null H0 : p = p0 = 0.25   Vs HA : p > 0.25 ANSWER 1

Part (b)

Sampling Distribution of X under H0 is: B(20, 0.25) ANSWER 2

Part (c)

P(Type I error)

= P(Rejecting H0 when it is in fact true)

= P(8 ? X ? 20/p = 0.25) [given in the question itself]

= 1 - P(X ? 7/p = 0.25)

= 1 – 0.898188

= 0.101812 ANSWER 3

Part (d)

P(Type II error)

= P(Accepting H0 when it is in fact not true)

= P(X ? 7/p = 0.3) [given in the question itself]

= 0.772272 ANSWER 4

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