In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. What is the 95% confidence interval for the true mean length of rods produced by this process?

95% Confidence interval: (14.4537 ,15.1463)

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 15 feet

Sample mean, = 14.8 feet

Sample size, n = 16

Alpha, α = 0.05

Sample standard deviation, σ = 0.65 feet

Degree of freedom = n - 1 = 15

95% Confidence interval:

Putting the values, we get,  

is the required confidence interval for the true mean length of rods.

95% confidence interval for the true mean length of rods produced by this process is [14.45 , 15.15].

Step-by-step explanation:

We are given that a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed.

A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet.

Firstly, the pivotal quantity for 95% confidence interval for the true mean length of rods is given by;

      P.Q. = ~

where, = sample mean length = 14.8 feet

             s = sample standard deviation = 0.65 feet

            n = sample of rods = 16

             = true mean

Here for constructing 95% confidence interval we have used t statistics because we don't know about population standard deviation.

So, 95% confidence interval for the population mean, is ;

P(-2.131 < < 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 < < 2.131) = 0.95

P( < < ) = 0.95

P( < < ) = 0.95

95% confidence interval for = [ ,  ]

                                             = [ , ]

                                             = [14.45 , 15.15]

Hence, 95% confidence interval for the true mean length of rods produced by this process is [14.45 , 15.15].

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