Question

The popular sport of mounting biking involves riding bikes off-road, often over very rough terrain. Trek is a leading manufacturer of mountain bikes, specially designed for durability and performance. A random sample of 10 Trek mountain bikes was obtained and each was carefully weighed. The resulting data yielded a sample mean of 10.67 kg and a sample standard deviation of 2.27 kg. Assume the underlying distribution of weights is normal. Find a 98% confidence interval for the true mean weight of Trek mountain bikes. (Round your answers to 2 decimal places.)

Answer 1

Answer: (8.64, 12.7)

Step-by-step explanation:

When population standard deviation is unknown then, confidence interval for population mean is given by :-

$\overline{x}\pm t_{(\alpha/2,\ df=n-1)}\dfrac{s}{\sqrt{n}}$

, where n= sample size , s= sample standard deviation, $\overline{x}$ = sample mean, $t_{\alpha/2, df=n-1}$ = two tailed t value for confidence level of 1-$\alpha$ and degree of freedom = 1-n.

Given: n= 10 , s= 2.27 kg , $\overline{x}=10.67$ kg

df = 10-1=9

For 98% confidence, significance level =$\alpha=1-0.98=0.02$

T-critical value: $t_{(0.02/2,9)}=t_{0.01,9}=2.8214\ \ \ [\text{By student's t-distribution table}]$

Now, 98% confidence interval for the true mean weight of Trek mountain bikes will be :

$10.67\pm (2.8214)\dfrac{2.27}{\sqrt{10}}\\\\=10.67\pm2.03\\\\ =(10.67-2.03,\ 10.67+2.03)\\\\=(8.64,\ 12.7)$

Hence, a 98% confidence interval for the true mean weight of Trek mountain bikes=  (8.64, 12.7)