An industrial designer wants to determine the average amount of time it takes an adult
to assemble an “easy to assemble” toy. A sample of 16 times yielded an average time
of 19.92 minutes, with a sample standard deviation of 5.73 minutes. Assuming normality
of assembly times, provide a 95% confidence interval for the mean assembly time.
Answers
95% confidence interval for the mean assembly time is [16.87 minutes , 22.97 minutes].
Step-by-step explanation:
We are given that an industrial designer wants to determine the average amount of time an adults take to assemble an "easy to assemble" toy.
A sample of 16 times yielded an average time of 19.92 minutes, with a sample standard deviation of 5.73 minutes.
Firstly, the Pivotal quantity for 95% confidence interval for the population mean is given by;
P.Q. = \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }
n
s
X
ˉ
−μ
~ t_n_-_1
where, \bar X
X
ˉ
= sample average time = 19.92 minutes
s = sample standard deviation = 5.73 minutes
n = sample size = 16
\muμ = population mean assembly time
Here for constructing 95% confidence interval we have used One-sample t-test statistics as we don't know about population standard deviation.
So, 95% confidence interval for the population mean, \muμ is ;
P(-2.131 < t_1_5 < 2.131) = 0.95 {As the critical value of t at 15 degree
of freedom are -2.131 & 2.131 with P = 2.5%}
P(-2.131 < \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }
n
s
X
ˉ
−μ
< 2.131) = 0.95
P( -2.131 \times {\frac{s}{\sqrt{n} } }−2.131×
n
s
< {\bar X-\mu}
X
ˉ
−μ < 2.131 \times {\frac{s}{\sqrt{n} } }2.131×
n
s
) = 0.95
P( \bar X-2.131 \times {\frac{s}{\sqrt{n} } }
X
ˉ
−2.131×
n
s
< \muμ < \bar X+2.131 \times {\frac{s}{\sqrt{n} } }
X
ˉ
+2.131×
n
s
) = 0.95
95% confidence interval for \muμ = [ \bar X-2.131 \times {\frac{s}{\sqrt{n} } }
X
ˉ
−2.131×
n
s
, \bar X+2.131 \times {\frac{s}{\sqrt{n} } }
X
ˉ
+2.131×
n
s
]
= [ 19.92-2.131 \times {\frac{5.73}{\sqrt{16} } }19.92−2.131×
16
5.73
, 19.92+2.131 \times {\frac{5.73}{\sqrt{16} } }19.92+2.131×
16
5.73
]
= [16.87 , 22.97]
Therefore, 95% confidence interval for the mean assembly time is [16.87 minutes , 22.97 minutes].
Now, the interpretation of the above confidence interval is that we are 95% confident that the mean assembly time will lie between 16.87 minutes and 22.97 minutes.