Question

2. Calcium levels in healthy adults are normally distributed with mean 9.5 mg/dl per deciliter and known standard devtion. A physician in a rural community suspects that the mean calcium levels are different for women i this comunity. He collects measurements from 20 healthy women and finds = 9.2 with sample standard deviation s 1.1. (a) (1 point) Describe the parameter of interest to the physician (b) (2 points) State the nl and alternative hypotheses (c)(1 point) Suppose we assume that the sample size here is large. Find a p-value for your test (d) (1 point) Does the resun (c) support his hypothesis? (e) (2 points) Given that the sample size is small, can you justify the use of a t-test in this situation? If you can, calculate a p-value for your test. If not, discuss why.

Answer 1

$(a)~\mu=population~mean~calcium~levels~for~women~in~a~rural~community\\ (b)~Null~hypothesis,~H_0:\mu=9.5\\ Alternative~hypothesis,~H_1:\mu\neq 9.5\\ (c)~Test~statistic=t=\frac{\sqrt{n}(\bar{x}-9.5)}{1.1}\\ Here~n=20,~\bar{x}=9.2\\ Value~of~t=-1.2197\\ p-value=2P(t>1.2197|t\sim t_{19})\\ Since~we~assume~that~the~sample~size~is~large~so~using~central~limit~theorem\\ t\overset{d}{\rightarrow}Normal(0,1)\\ Hence~p-value=2(1-\Phi(1.2197))=0.2226\\ (d)~Since~p-value>0.05~so~we~fail~to~reject~H_0~at~5\%~level~of~significance\\ and~conclude~that~there~is~insufficient~evidence~to~support~his~claim.\\ (e)~Since~the~ normal~ distribution~ has ~relatively~ more~ scores ~in~ the ~center ~of~ the ~distribution\\ and ~the~ t ~distribution~ has ~relatively ~more ~in ~the~ tails~so~if~we~use\\ ~t-distribution~with~19~df~in~this~situation~then~we~get~larger~p-value~than\\ previous.~Since~we~accept~H_0~using~normal~distribution~so~here~also~we~accept~H_0.\\$$So~we~do~not~need~to~compute~p-value~in~this~situation.$