A soft drink producing company knew that the proportion of their under- filled b

Question

A soft drink producing company knew that the proportion of their under- filled bottles was 11%. The old bottle filling production line was replaced by a new one in an attempt to reduce the old proportion of under-filled bottles. A random sample of 120 bottles were selected from the new filling line and it turns out that 12 of them were under-filled. [2] a. Set the appropriate null and alternative hypotheses to test if the proportion of under-filled bottles has decreased. [4] What does the TYPE I error mean in the context of this example. Explain. [6] c. Compute the value of the appropriate test statistic for the hypothesis testing in part a. [4] d. Compute the p-value for the hypothesis testing problem in part a based on the value of the test statistic computed in part c. and USING the p-value draw a conclusion at level alpha = 0.05.

Explanation / Answer

a. Here p = proportion of under filled bottles

p0= old bottle filling production line

p = New bottle filling production line

Null Hypothesis : H0 : There is no change in under filled bottles proportion with respect to change in production line ( p = p0)

Alternative Hypothesis :H1: There is significant reduction in under filled bottles proportion with respect to change in production line ( p< p0)

(b) Type I error in this context mean that there is no difference between under filled bottles proportion with respect to different production lines but due to any condition ( may be false measurement etc.), there are significantly difference is detected between production lines.

(c) Test statistic : Here p0 = 0.11

and p = 12/120 = 0.10

and standard deviation s= sqrt [p0( 1- p0)/n] = sqrt [ 0.11 * 0.89 / 120] = 0.0286

z - statistic = ( p - p0 )/ s = ( 0.10 - 0.11)/ 0.0286 = - 0.35

p - value = Prob ( pbar < 0.10 if p = 0.11) = Prob ( Z < -0.35) = 0.3632

so, p - value is greater than significance level alpha = 0.05

so we cannot reject the null hypothesis and can say there is no effect of under filled bottles proportion with respect to new and old production lines.

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