The owner of Kat Motel wants to develop a time standard for the task of cleaning a cat cage. In a preliminary study, she observed one of her workers perform this task six times, with the following results: $$\begin{array} { | l | c | c | c | c | c | c | }
\hline \text { Observation } & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \text { Time(Seconds) } & 99 & 87 & 90 & 81 & 93 & 90 \
\hline
\end{array}$$ How many observations should be made if she wants to be 95.44 percent confident that the maximum error in the observed time is two seconds? Assume that the standard deviation of the task time is six seconds.
A) 5
B) 6
C) 30
D) 36
E) 49

Question

The owner of Kat Motel wants to develop a time standard for the task of cleaning a cat cage. In a preliminary study, she observed one of her workers perform this task six times, with the following results: $$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Observation } & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \text { Time(Seconds) } & 99 & 87 & 90 & 81 & 93 & 90 \
\hline
\end{array}$$ How many observations should be made if she wants to be 95.44 percent confident that the maximum error in the observed time is two seconds? Assume that the standard deviation of the task time is six seconds.
A) 5
B) 6
C) 30
D) 36
E) 49

Quizplus · Accepted Answer

To determine the sample size needed, we use the formula:

$n=\left(\dfrac{z\sigma}{E}\right)^2$

where:
z = the z-score associated with the confidence level (95.44% confidence level = 1.81)
$\sigma$ = the population standard deviation (6 seconds)
E = the maximum error in the observed time (2 seconds)

Plugging in the values, we get:

$n=\left(\dfrac{1.81\cdot6}{2}\right)^2=36.4692$

Rounding up to the nearest whole number, we get:

$n=37$

Therefore, we need to make 37 observations to be 95.44% confident that the maximum error in the observed time is two seconds. However, since we already have 6 observations, we only need to make:

$n-6=37-6=31$

more observations. Therefore, the correct answer is (C) 30, as it is the closest option to 31.

The Owner of Kat Motel Wants to Develop a Time