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poizon [28]
3 years ago
9

In a sample of 60 electric motors, the average efficiency (in percent) was 85 and the standard deviation was 2. Section 05.01 Ex

ercise 12.a - Compute confidence interval; Find necessary sample size Find a 95% confidence interval for the mean efficiency. Round the answers to three decimal places.
Mathematics
1 answer:
jasenka [17]3 years ago
8 0

Answer:

95% confidence interval for the mean efficiency is [84.483 , 85.517].

Step-by-step explanation:

We are given that in a sample of 60 electric motors, the average efficiency (in percent) was 85 and the standard deviation was 2.

So, the pivotal quantity for 95% confidence interval for the population mean efficiency is given by;

          P.Q. = \frac{\bar X - \mu}{\frac{s}{\sqrt{n} } } ~ t_n_-_1

where, \mu = sample average efficiency = 85

            \sigma = sample standard deviation = 2

            n = sample of motors = 60

            \mu = population mean efficiency

<em>So, 95% confidence interval for the mean efficiency, </em>\mu<em> is ;</em>

P(-2.0009 < t_5_9 < 2.0009) = 0.95

P(-2.0009  < \frac{\bar X - \mu}{\frac{s}{\sqrt{n} } } < 2.0009 ) = 0.95

P( -2.0009  \times {\frac{s}{\sqrt{n} } < {\bar X - \mu} < 2.0009  \times {\frac{s}{\sqrt{n} } ) = 0.95

P( \bar X -2.0009  \times {\frac{s}{\sqrt{n} } < \mu < \bar X +2.0009  \times {\frac{s}{\sqrt{n} } ) = 0.95

<u>95% confidence interval for</u> \mu = [ \bar X -2.0009  \times {\frac{s}{\sqrt{n} } , \bar X +2.0009  \times {\frac{s}{\sqrt{n} } ]

                                                 = [ 85 -2.0009  \times {\frac{2}{\sqrt{60} } , 85 +2.0009  \times {\frac{2}{\sqrt{60} } ]

                                                 = [84.483 , 85.517]

Therefore, 95% confidence interval for the population mean efficiency is [84.483 , 85.517].

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