Many researchers claim that the mean serum indirect bilirubin level of 4-day-old

Question

Many researchers claim that the mean serum indirect bilirubin level of 4-day-old infants is 6 mg/dl. We wish to test if it’s really 6 mg/dl or not. We collect the sample of 16 4-day-old infants and find the mean serum to be 5.98 mg/dl. Assuming bilirubin levels of 4-day-old infants are approximately normal distributed with a standard deviation of 3.5 mg/dl:

a) Find and interpret 90% confidence interval for the true mean bilirubin level.

b) Conduct the hypothesis at = 0.1.

c) Find and interpret 95% confidence interval for the true mean bilirubin level.

d) Conduct the hypothesis at = 0.05.

e) Compare the conclusions you draw from (a) and (b). Similarly, from (c) and (d).

Explanation / Answer

Answer to part a)

x bar = 5.98

n = 16

= 3.5

Formula of Confidence interval is :

x bar - Z * / n , x bar + Z * / n

.
For 90% confidence level , Z = 1.28

.

On plugging the values weget:

5.98 - 1.28 * 3.5 /sqrt(16) , 5.98 + 1.28 *3.5 /sqrt(16)

5.98 -1.12 , 5.98 +1.12

4.86 , 7.10

.

Answer to part b)

The hypothesis for this test are as follows:

Ho: M = 6

Ha: M 6

[It is a two tailed test]

.

For conducting hypothesis test we need to find the Test Statistic

Z = ( xbar - M) / ( / n )

Z = (5.98 -6) / (3.5/16)

Z = -0.02 / 0.875

Z = -0.02

.

The P value for this is : 0.4920

.

Inference since the P value 0.4920 > 0.01 , we fail to reject the null hypothesis

Conclusion: The mean is equal to 6

.

Answer to part c)

For 95% interval Z = 1.96

5.98 -1.96 *3.5/sqrt(16) , 5.98 +1.96 *3.5 /sqrt(16)

5.98 -1.715 , 5.98 +1.715

4.265 , 7.695

.

This means when various samples are taken , 95% of the samples have the true population mean in the range 4.265 to 7.695

.

Answer to part d)

We already got the Z = -0.02

and P value = 0.4920

.

Inference : Since P value > 0.05 , we fail to reject the null

Conclusion: Thus the mean is 6

.

Answer to part e)

As per part a confidence interval is 4.86 , 7.10 , 6 lies in this interval , adn that is what part b also says , that 6 is the true population mean

similarly , as per part c the confidence interval is 4.265 to 7.695, 6 lies in this interval , and that is what part dalso says, that 6 is the true population mean

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