Question

A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered.

z = 1.33

p- value = 0.0913

(b) If it is really the case that 16% of all plates blister under these circumstances and a sample size 100 is used, how likely is it that the null hypothesis of part (a) will not be rejected by the 0.05 test? (Round your answer to four decimal places.)

If it is really the case that 16% of all plates blister under these circumstances and a sample size 200 is used, how likely is it that the null hypothesis of part (a) will not be rejected by the 0.05 test? (Round your answer to four decimal places.)

(c) How many plates would have to be tested to have β(0.16) = 0.10 for the test of part (a)? (Round your answer up to the next whole number.)

Answer 1

Given that z = 1.33, and p value is 0.0913, n = 100

By plugging in the values and solving in the equation

we get p = 0.10 and the p value is for a right tail, hence this is a right tailed test with hypothesis

H0: p = 0.10

Ha: p > 0.10

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(a) To find the probability that the null hypothesis will not be rejected (when it should be), i.e probability of a Type II error,

= 0.05 and the z critical value (right tailed) is 1.645

Given that the true proportion is 16% = 0.16

The value of at which the null gets rejected

Solving, we get = 0.149

When the true proportion is 0.16, then P( < 0.149) = P(type II error)

The probability of a Type II error (from the normal tables) = 0.3859

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n = 200, to find the probability of a Type II error

The value of at which the null gets rejected

Solving, we get = 0.135

When the true proportion is 0.16, then P( < 0.135) = P(type II error)

The probability of a Type II error (from the normal tables) = 0.1660

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(c) Sample size when power of the test, = 0.10

Hypothesis proportion p0 = 0.10 and true proportion p1 = 0.16, = 0.05

Zcritical (alpha0 = 1.645 and Z critical (Beta) = 1.282

The calculation for sample size n is given by

n = 258