Question

Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution. 10.1 9.4 10.1 9.5 9.4 9.8 10.0 9.9 11.2 12.1 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean reading and the sample standard deviation s. (in mg/dl; round your answers to two decimal places.) = mg/dl s = mg/dl (b) Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (in mg/dl; round your answer to two decimal places.) lower limit mg/dl upper limit mg/dl (c) Based on your results in part (b), do you think this patient still has a calcium deficiency? Explain. Yes. This confidence interval suggests that the patient may still have a calcium deficiency. Yes. This confidence interval suggests that the patient no longer has a calcium deficiency. No. This confidence interval suggests that the patient may still have a calcium deficiency. No. This confidence interval suggests that the patient no longer has a calcium deficiency.

Answer 1

Solution

Part (a)

Mean = 10.09 mg/dl Answer 1

Standard deviation = 0.84 mg/dl Answer 2

Part (b)

100(1 - α) % Confidence Interval for population mean μ, when σ is not known is: Xbar ± MoE

where

MoE = (t_{n- 1, α /2})s/√n

with

Xbar = sample mean,

t_{n – 1, α /2} = upper (α/2)% point of t-distribution with (n - 1) degrees of freedom,

s = sample standard deviation and

n = sample size.

So, 99.9% confidence interval for the population mean of total calcium in the patient's blood

Lower limit = 8.82 mg/dl and Upper limit = 11.36 mg/dl Answer 3

Details of calculations

Given	α	0.001	1 - (α/2) =	0.9995
	n	10	SQRT(n)	3.16227766
	Xbar	10.0909	n - 1	9
	s	0.8396
tα/2		4.780912586
99% CI for μ		10.0909	±	1.26935539
Lower Bound		8.821544605
Upper Bound		11.36025539

Part (c)

No. This confidence interval suggests that the patient no longer has a calcium deficiency. Answer 4

[The confidence interval does not contain the stipulated standard of ‘calcium level below 6 mg/dl’ to be declared calcium deficiency.]

DONE