Question

The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Let x₁ be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952. Based on 9160 observations, the sample mean interval was x₁ = 62.8 minutes. Let x₂ be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987. Based on 25,457 observations, the sample mean time interval was x₂ = 72.2 minutes. Historical data suggest that σ₁ = 8.84 minutes and σ₂ = 11.85 minutes. Let μ₁ be the population mean of x₁ and let μ₂ be the population mean of x₂.

(a) Compute a 99% confidence interval for μ₁ – μ₂. (Use 2 decimal places.)

lower limit
upper limit

Answer 1

SOLUTION:

From given data,

$\bar{x}_{1}$ =62.8	$\bar{x}_{2}$ = 72.2
01 =8.84	$\sigma _{2}$ = 11.85
= 9160	= 25457

(a) Compute a 99% confidence interval for μ₁ – μ₂. (Use 2 decimal places.)

99% confidence interval

Confidence interval is 99%

99% = 99/100 = 0.99

$\alpha$ = 1 - Confidence interval = 1-0.99 = 0.01

$\alpha$/2 = 0.01 / 2

= 0.005

Z_$\alpha$/2 = Z_0.005  = 2.33

($\bar{x}_{1}$-$\bar{x}_{2}$) $\pm$ Z_$\alpha$/2 * sqrt($\sigma_{1}^{2}$/ + $\sigma_{2}^{2}$ / )  

($\bar{x}_{1}$-$\bar{x}_{2}$) - Z_$\alpha$/2 * sqrt($\sigma_{1}^{2}$/ + $\sigma_{2}^{2}$ / ​​​​​​​​​​​​​​ ) <  $\mu _{1}$-$\mu _{2}$ <  ($\bar{x}_{1}$-$\bar{x}_{2}$) + Z_$\alpha$/2 * sqrt($\sigma_{1}^{2}$/ + $\sigma_{2}^{2}$ / ​​​​​​​​​​​​​​ )

(62.8 -72.2 ) - 2.33 * sqrt(8.84² / 9160  + 11.85² / 25457) <  $\mu _{1}$-$\mu _{2}$ < (62.8 -72.2 ) + 2.33 * sqrt(8.84² / 9160  + 11.85² / 25457)

-9.4- 2.33 * 0.11852107 <  $\mu _{1}$-$\mu _{2}$ < -9.4+ 2.33 * 0.11852107

-9.4- 0.2761540931 <  $\mu _{1}$-$\mu _{2}$ < -9.4+ 0.2761540931

-9.676154 <  $\mu _{1}$-$\mu _{2}$ < -9.1238459

- 9.68 <  $\mu _{1}$-$\mu _{2}$ < - 9.12

lower limit : - 9.68 minutes

upper limit: - 9.12 minutes

Please thumbs-up / vote up this answer if it was helpful. In case of any problem, please comment below. I will surely help. Down-votes are permanent and not notified to us, so we can't help in that case.