Question

Acrylic bone cement is commonly used in total joint replacement to secure the artificial joint. Data on the force (measured in Newtons, N) required to break a cement bond was determined under two different temperature conditions and in two different mediums appear in the following table. Temperature Medium Data on Breaking Force 22 degrees Dry 100.6, 142.7, 194.8, 118.4, 176.1, 213.1 37 degrees Dry 302.3, 339.3, 288.8, 306.8, 305.2, 327.5 22 degrees Wet 385.3, 368.2, 322.6, 307.4, 357.9, 321.4 37 degrees Wet 363.9, 376.5, 327.7, 331.9, 338.1, 394.6 (a) Estimate the difference between the mean breaking force in a dry medium at 37 degrees and the mean breaking force at the same temperature in a wet medium using a 90% confidence interval. (Round your answers to one decimal place.) ( , ) (b) Is there sufficient evidence to conclude that the mean breaking force in a dry medium at the higher temperature is greater than the mean breaking force at the lower temperature by more than 100 N? Test the relevant hypotheses using a significance level of 0.10. (Use μhigher temperature − μlower temperature. Round your test statistic to two decimal places. Round your degrees of freedom to the nearest whole number. Round your p-value to three decimal places.) t = df = P = Conclusion Yes, there is sufficient evidence. No, there is not sufficient evidence.

Answer 1

(a) The 90% confidence interval for mean breaking force in a dry medium at 37 degress and wet medium at 37 degrees is given by

where

: population mean breaking force in a dry medium at 37 degress

: population mean breaking force in a wet medium at 37 degress

= 23.1589

degrees of freedom = n1+n2-2 = 10

For 90% confidence with df = 10 , critical value of t is

tc = 1.81 ( from t table)

Thus, 90% confidence interval for mean braeking force in a dry medium at 37 degress and wet medium at 37 degrees is given by

= ( -68.0 , - 19.6)

(b) The null and alternative hypothesis is

where

mean breaking force in a dry medium at higher temperature (37 degree)

mean breaking force in a dry medium at lower temperature (22 degree)

Test statistic

Test statistic is

= 2.76

degrees of freedom calculation

= 147057

=22038.19

df = 6.7 =7

For 0.01 level of significance with df =7 , one tailed critical value of t is

tc =1.41 ( from t table)

Since t > 1.41

We reject H0

YES, There is sufficient evidence to conclude that mean breaking strength in a dry medium at higher temperature is greater  than mean braeking strength in a dry medium at lower temperature by more than 100N .

Note : For part (a) we used the formula for equal varinace

But in part (b) , as the sample variances seems highly different from each other , we used t test formula for unequal variance

Formula for sample mean and variance

and