Question

Here are summary statistics for randomly selected weights of newborn​ girls: nequals177​, x overbarequals28.9 ​hg, sequals6.7 hg. Construct a confidence interval estimate of the mean. Use a 98​% confidence level. Are these results very different from the confidence interval 28.1 hgless thanmuless than30.7 hg with only 20 sample​ values, x overbarequals29.4 ​hg, and sequals2.3 ​hg? What is the confidence interval for the population mean mu​?

Answer 1

Question:

Here are summary statistics for randomly selected weights of newborn​ girls: n = 177​, x = 28.9 ​hg, s = 6.7 hg. Construct a confidence interval estimate of the mean. Use a 98​% confidence level. Are these results very different from the confidence interval 28.1 hg < μ < 30.7 hg with only 20 sample​ values, x' =29.4 ​hg, and s = 2.3 ​hg? What is the confidence interval for the population mean μ​?

Answer:

The confidence interval for the population mean μ is 27.73 ≤ μ ≤ 30.0714

Explanation:

The equation to identify the confidence interval for the mean is given by

`x'-z_{(\alpha )/(2)} (s)/(√(n) ) \leq \mu\leq x'+z_{(\alpha )/(2)} (s)/(√(n) )`

Where

x' = Sample mean = 28.9

s = Standard deviation = 6.7

n = Sample size = 177

`z_{(\alpha )/(2)}` = Critical value = 2.326

Therefore we have

`28.9-2.326(6.7)/(√(177) ) \leq \mu\leq 28.9+2.326 (6.7)/(√(177) )`

27.73 ≤ μ ≤ 30.0714

T test we have

t =
`(x'-\mu)/((s)/(√(n) ) )`

=
`(29.4-28.9)/((2.3)/(√(16) ) )` = 0.8696 which is < 1

df = 15 as sample size = 15

Upper tail statistics lies between 0.3 and 0.1

Answer 2

(32.2,34.7)

Explanation:

Solution :

Given that,

\bar x = 33.4 ​hg

s = 6.4 hg

n = 177​

Degrees of freedom = df = n - 1 = 177 - 1 = 176

At 99% confidence level the t is ,

α = 1 - 99% = 1 - 0.99 = 0.01

α / 2 = 0.01 / 2 = 0.005

tα /2,df = t0.005,176 = 2.604

Margin of error = E = tα/2,df * (s /√n)

= 2.604* ( 6.4/ √177)

= 1.25

The 95% confidence interval estimate of the population mean is,

\bar x - E < \mu < \bar x + E = 33.4 - 1.25 < \mu < 33.4 + 1.25

32.15 < \mu < 34.65

32.2 < \mu < 34.7

(32.2,34.7)