Question

An insurance company claims the average car on the road is less than 6 years old. Based on a random sample of 15 cars, the mean age is 5.8 years with a standard deviation of 1.1 years. Does the sample support the insurance company's claim?

Answer 1

Answer: No, we do not have enough evidence to  support the insurance company's claim.

Step-by-step explanation:

Let $\mu$ be the average car on the road .

As per given , we have

Null hypothesis : $H_0:\mu\geq6$

Alternative hypothesis : $H_a:\mu<6$

Since ,  $H_a$ is left-talied and population standard deviation is unknown , so we perform a left-tailed t-test.

Test statistic : $t=\dfrac{\overline{x}-\mu}{\dfrac{s}{\sqrt{n}}}$ , where $\overline{x}$= sample mean , s= sample standard deviation , n= sample size .

Put  $\overline{x}$=  5.8 years , s=  1.1 years , n= 15 .

$t=\dfrac{5.8-6}{\dfrac{1.1}{\sqrt{15}}}\approx-0.704$

Also, At 0.05 significance ,

 $t_{critical}=1.75305$   (by t-distribution table)

Decision : Since $|t_{calculated}|<|t_{critical}| \ [\ \because\ 0.704<1.75305]$ , so we fail to reject the null hypothesis .

Conclusion : At 5% confidence level , we do not have enough evidence to  support the insurance company's claim.