A study was conducted to determine standard reference values for musculoskeletal

Question

A study was conducted to determine standard reference values for musculoskeletal ultrasonography in healthy adults. For an independent random sample of 54 women, sagittal diameter (in mm) of the biceps tendon was taken and resulted in sample statistics y = 2.42 and s = 0.48. (a) Construct a 75% confidence interval for mu, the population average of sagittal diameter of the biceps tendon. State whether this interval is exact or approximate and motivate your answer. (b) For the observed sample statistics, determine the sum of the data values as well as the sum of the squared data values, i.e. sigma_i=1^54 y_i and sigma_i=1^54 y_i^2 (c) An additional three measurements are obtained, namely y_55 = 2.83, y_56 = 2.95 and y_57 = 1.34. Calculate the new sample mean and sample standard deviation. Also calculate a new 75% confidence interval for mu making use of the data. Discuss the length of the CI before and after new data was added. Specifically, refer to the lengths of the intervals and comment on these.

Explanation / Answer

Back-up Theory

Let Y = Sagittal diameter of biceps tendon (in mm). We assume Y ~ N(µ, 2).

Now, to work out solution,

Part (a)

100(1 – ) % confidence interval for µ when 2 is unknown is: {Ybar ± (s/n)(t/2)}, where

Ybar = sample mean,

= population standard deviation,

s = sample standard deviation,

n = sample size and

t/2 = upper (/2) % point of t-Distribution with (n - 1) degrees of freedom..

Given, n = 54, = 0.25, Ybar = 2.42, s = 0.48, and

t/2 = t53,0.125 = 1.5588, [using Excel Function],

75% Confidence Interval for µ is: {2.42 ± (0.48/54)(1.5588)} = (2.42 ± 0.1018 )

Lower Bound = 2.318, Upper Bound = 2.522 ANSWER

Part (b)

Ybar = {[1,54]yi}54 and hence [1,54]yi = 54 x 2.42 = 130.68 ANSWER 1

s2 = {[1,54](yi – Ybar)2}/53 = {[1,54](yi2) – 54xYbar2}/53 or

[1,54](yi2) = (53x0.482) + (54x2.422) = 328.4568 ANSWER 2

Part (c)

For 54 values. [1,54]yi = 130.68 [1,54](yi2) = 328.4568

When 3 new values are added,

For 57 values. [1,54]yi = 130.68 + 2.83 + 2.95 + 1.34 = 137.8 and

[1,54](yi2) = 328.4568 + 2.832+ 2.952+ 1.342= 346.964

So, the new values are:

Ybar = 328.4568/57 = 2.4175 and

s2 = {346.964 - (57x2.41752)}/56 = (346.964 - 333.1255)/56 = 13.83834/56 = 0.247113281 and hence s = 0.497105

Thus, the new values are: mean 2,418 and s = 0.497 ANSWER 1

75% Confidence Interval for µ is: {2.418 ± (0.497/57)(1.5574)} = (2.418 ± 0.1025 )

Lower Bound = 2.316, Upper Bound = 2.521 ANSWER 2

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.