Question

A big-toe problem 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 dotplot of the data revealed no outliers or strong skewness. $^{27}$ (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.

Answer 1

A 90% confidence interval for the mean HAV angle for the 21 patients is approximately 22.37 to 27.15 degrees. Including an outlier of 50 degrees would increase both the mean and standard deviation, resulting in a wider and less precise confidence interval shifted upwards.

Step-by-step explanation:

To construct a 90% confidence interval for the mean HAV angle in the population of all such patients, we use the sample mean, standard deviation, and the sample size. Since the sample size is relatively small (21 patients), we use the t-distribution for the confidence interval calculation.

The formula for the confidence interval is:

Mean ± (t-value × (Standard Deviation / √Sample Size))

Where the t-value is determined by the degrees of freedom (n - 1) and the desired level of confidence. Here, the degrees of freedom are 20 (21-1).

Using a t-distribution table or software to find the t-value for 20 degrees of freedom at a 90% confidence level, we get approximately 1.725. Now, we can calculate the confidence interval:

24.76 ± (1.725 × (6.34 / √21))

This results in a confidence interval of approximately:

24.76 ± 2.39

So, the 90% confidence interval for the mean HAV angle is about 22.37 to 27.15 degrees.

b) If we include the outlier of 50 degrees, it would increase the variability (standard deviation) of the data. This increased variability leads to a wider confidence interval. The mean would also shift higher due to the larger value of the outlier, leading to a confidence interval that is shifted upwards and less precise.

Answer 2

Alright. So on the sample size and it's 21 patients. Okay. And the sample means HIV angle X. Bar is 84.76 degrees. Okay and the sample standard deviation S. T. D. Is 6.34 degrees. Now in part A we have to construct 90% carpenters interval part the mean HIV angle in the population of all such patients. Okay, so the Pamela Pardy had confidence in Turnbull is X. R. Plus. Manistee is of critical times sigma over square root N. Okay, So using the T critical value calculator at a 90% confidence interval in degrees of freedom is In -1 which is 21 -1. This is 20 part true tale gives it the critical value tease of C Is 1.7847. Okay so the 90% confidence in terrible is 24.76 plus men as 1.7247 times six point three Poor Divided by Square Root. 21. Okay so this will be equal to 24.76 plus -2.8861. Uh this is equal to 24.76 -2.3861 and 24.76 plus 2.3861. Okay and this is equal to 22.3739 and 27.1461. Okay. Now the interpreter and there is a 90% confidence interval that mean HIV angle patients population is lies between 22.3739 degrees and 27.1 for uh 61. Okay, so in part B The regression admitted one patient within HIV Angola, 50° from the analysis. So 50° is very far from mean 24.76° as compared to the standard deviation, Which is 6.3 point okay. Now the outliers uh effect the mean and uh the our standard deviation. So the values are meaning at the standard deviation are uh were valued when extreme values are present. Okay, so if we remove large valued outlier with value pip pip from sample, uh then mean and standard deviation. Both are decreased below to effect, including this our clear on the competence center. Well, okay, So let here, in part one The the upper and lower limits are higher when this outlier is present as compared to the upper and lower limits without outlier. And uh and part two Ah and by two ah with the competence in turbo is large when this our clear is prison but with his butt with is increased. I went out there does not include in data. Okay.