Chapter 8: Q. 68 (page 519)

Hallux abducto valgus (call it HAV) is a deformation of the big toe that is fairly uncommon in youth and often requires surgery. Doctors used $X$ -rays to measure the angle (in degrees) of deformity in a random sample of patients under the age of $21$ who came to a medical center for surgery to correct HAV. The angle is a measure of the seriousness of the deformity. For these $21$ patients, the mean HAV angle was $24.76$ degrees and the standard deviation was $6.34$ degrees. A dot plot of the data revealed no outliers or strong skewness.
(a) Construct and interpret a $90 %$ confidence interval for the mean HAV angle in the population of all such patients.
(b) Researchers omitted one patient with an HAV angle of $50$ degrees from the analysis due to a measurement issue. What effect would including this outlier have on the confidence interval in (a)? Justify your answer.

Short Answer

Expert verified

(a) We are $90 %$ confident that the true population mean is between $22.3735$ degrees and $27.1465$ degrees.

(b) The confidence interval from part (a) could also not be generalized for the population, because the normal requirement for the use of the $t$ -distribution is not satisfied.

Step by step solution

Part (a) Step 1: Given Information

Given

$\bar{x} = 24.76$

$s = 6.34$

$n = 21$

$c = 90 %$

Part (b) Step 2: Explanation

Determine the t-value by looking in the row starting with degrees of freedom and in the row with $c = 90 %$ in table B :

$t^{*} = 1.725$

The margin of error is then:

localid="1650086939662" $E = t^{*} \cdot \frac{s}{\sqrt{n}} = 1.725 \times \frac{6.34}{\sqrt{21}} \approx 2.3865$

Then the confidence interval becomes:

localid="1650086961671" $22.3735 = 24.76 - 2.3865 = \bar{x} - E < μ < \bar{x} + E = 24.76 + 2.3865 = 27.1465$

$22.3735 < μ < 27.1465$