Question

A toy maker claims his best product has an average lifespan of exactly 18 years. A skeptical product evaluator asks for evidence (data) that might be used to evaluate this claim. The product evaluator was provided data collected from a random sample of 35 people who used the product. Using the data, an average product lifespan of 13 years and a standard deviation of 2 years was calculated. Select the 99%, confidence interval for the true mean lifespan of this product.

Answer 1

99%, confidence interval for the true mean lifespan of this product is [12.08 years , 13.92 years].

Explanation:

We are given that a toy maker claims his best product has an average lifespan of exactly 18 years.

The product evaluator was provided data collected from a random sample of 35 people who used the product. Using the data, an average product lifespan of 13 years and a standard deviation of 2 years was calculated.

Firstly, the pivotal quantity for 99% confidence interval for the true mean is given by;

P.Q. =
`(\bar X-\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample average product lifespan = 13 years

n = sample of people = 35

s = sample standard deviation = 2 years

`\mu` = true mean lifespan

Here for constructing 99% confidence interval we have used One-sample t statistics because we don't know about the population standard deviation.

So, 99% confidence interval for the true mean,
`\mu` is ;

P(-2.728 <
`t_3_4` < 2.728) = 0.99 {As the critical value of t at 34 degree

of freedom are -2.728 & 2.728 with P = 0.5%}

P(-2.728 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 2.728) = 0.99

P(
`-2.728 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`2.728 * {(s)/(√(n) ) }` ) = 0.99

P(
`\bar X-2.728 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+2.728 * {(s)/(√(n) ) }` ) = 0.99

99% confidence interval for
`\mu` = [
`\bar X-2.728 * {(s)/(√(n) ) }` ,
`\bar X+2.728 * {(s)/(√(n) ) }`]

= [
`13-2.728 * {(2)/(√(35) ) }` ,
`13+2.728 * {(2)/(√(35) ) }` ]

= [12.08 , 13.92]

Therefore, 99% confidence interval for the true mean lifespan of this product is [12.08 years , 13.92 years].