Question

2- A sample of 49 sudden infant death syndrome (SIDS) cases had a mean birth weight of 2998 g. Based on other births in the county, we will assume ơ 800g. Calculate the 95% confidence interval for the mean birth weight of SIDS cases in the county. Interpret your results.

Answer 1

Solution:

Confidence Interval for population mean using z distribution

Given,

$\bar x$ = 2998 ....... Sample mean

$\sigma$   = 800 ........Population standard deviation

n = 49 ....... Sample size

Note that, Population standard deviation($\sigma$) is known..So we use z distribution.

Our aim is to construct 95% confidence interval.

$\therefore$ c = 0.95

$\therefore$$\alpha$ = 1- c = 1- 0.95 = 0.05

$\therefore$  $\alpha$/2 = 0.05 $\slash$ 2 = 0.025 and 1- $\alpha$ /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

$\therefore$$Z_{\alpha/2}$ = 1.96 for the 95% Confidence interval

Now , confidence interval for mean($\mu$) is given by:

$\bar x - Margin. of .error < \mu < \bar x + Margin. of .error$

  $\bar x-Z_{\alpha/2}*(\sigma/\sqrt{n}) < \mu < \bar x+Z_{\alpha/2}*(\sigma/\sqrt{n})$

2998 -1.96*(800/ $\sqrt{}$ 49)   $< \mu <$    2998 +1.96*(800/ $\sqrt{}$ 49)

2998-224 < $\mu$ <   2998+224

2774 < $\mu$ < 3222

So, (2774 , 3222) is the required 95% confidence interval for mean.

95% C.I. means that there is 0.95 probability that the true mean (population mean) will lie within the interval.