Confidence Interval Probabilty1) We have discovered a new method to produce a liquid containing Zinc. We measure the concentration μ four times independently (from the same solution) and find the values0.3, 0.33, 0.3, 0.27a) Assuming that the standard deviation of the measurement error is 0.02 (when we make one measurement) estimate μ and give a 90%-confidence interval.Answer:We assume the measurement errors to be normal with zero expectation. The estimate for μ is μ X = ( X 1 + X 2 + X 3 + X 4 ) / 4 = ( 0.3 + 0.33 + 0.3 + 0.27 ) / 4 = 0.3.The same procedure as usual leads to a confidence interval [ 0.3 − b standard deviation / n , 0.3 + b standard deviation / n ]where the standard deviation equals 0.02 and n = 2, whilst b is defined by the equation P ( − b ≤ N ( 0 , 1 ) ≤ b ) = 90 % .The last equation above is equivalent to λ ( b ) = 0.95 which we can solve with the help of a table and find b = 1.645. Hence the 90% confidence interval is equal to [ 0.3 − 0.0164 , 0.3 + 0.0164 ] = [ 0.2836 , 0.3164 ]. The true concentration is thus between 0.2836 and 0.3164 with at least 95% probability.My question is how were they able to get 95% probability and b?

Question

Confidence Interval Probabilty1) We have discovered a new method to produce a liquid containing Zinc. We measure the concentration   μ four times independently (from the same solution) and find the values0.3, 0.33, 0.3, 0.27a) Assuming that the standard deviation of the measurement error is 0.02 (when we make one measurement) estimate μ and give a 90%-confidence interval.Answer:We assume the measurement errors to be normal with zero expectation. The estimate for   μ is            μ        X    =  (      X    1    +      X    2    +      X    3    +      X    4    )      /    4  =  (  0.3  +  0.33  +  0.3  +  0.27  )      /    4  =  0.3.The same procedure as usual leads to a confidence interval  [  0.3  −      b    standard deviation        /        n    ,    0.3  +      b    standard deviation        /        n    ]where the standard deviation equals 0.02 and       n    =  2, whilst b is defined by the equation  P  (  −  b  ≤  N  (  0  ,  1  )  ≤  b  )  =  90  %  .The last equation above is equivalent to   λ  (  b  )  =  0.95 which we can solve with the help of a table and find   b  =  1.645. Hence the 90% confidence interval is equal to   [  0.3  −  0.0164  ,  0.3  +  0.0164  ]  =  [  0.2836  ,  0.3164  ]. The true concentration is thus between 0.2836 and 0.3164 with at least 95% probability.My question is how were they able to get 95% probability and b?

Jean Deleon · Accepted Answer

Step 1You want to find b such that   P  (  −  b  ≤  N  (  0  ,  1  )  ≤  b  )  =  0.9. From the symmetry of the normal density function, this means that  P  (  b  &amp;lt;  N  (  0  ,  1  )  &amp;lt;  ∞  )  =  P  (  −  ∞  &amp;lt;  N  (  0  ,  1  )  &amp;lt;  b  )  =  0.05since 10% of the area under the normal density function is outside the interval [-b,b]. You have a table of values of the cumulative standard normal distribution function, which tells you the value of  P  (  −  ∞  &amp;lt;  N  (  0  ,  1  )  ≤  b  )  =  P  (  −  ∞  &amp;lt;  N  (  0  ,  1  )  &amp;lt;  b  )  +  P  (  −  b  ≤  N  (  0  ,  1  )  ≤  b  )  =  0.05  +  0.9  =  0.95So you look in the table and find that   b  =  1.645 gives a value of 0.9495, just a little less than the desired 0.9500, and so the exact value of b is a tad more that 1.645.

Jaylyn Horne · Accepted Answer

Step 1I&#039;m not fully certain what the question is, but for I&#039;ll guess that it&#039;s something like this: The probability that a normally distributed random variable is less than its expected value plus 1.645 standard deviations is about 95%, so how is that related to the 90% figure mentioned after that?Step 2  Pr  (  Z  &amp;lt;  1.645  )  =  0.95.So  Pr  (  Z  &amp;gt;  1.645  )  =  0.05.By symmetry, then,  Pr  (  Z  &amp;lt;  −  1.645  )  =  0.05.So you have 5% of the probability in the upper tail and 5% in the lower tail. That makes 10%. So 90% remains in the middle.

1) We have discovered a new method to produce a liquid containing Zinc. We measure the concentration μ four times independently (from the same solution) and find the values 0.3, 0.33, 0.3, 0.27.

Answered question

Answer & Explanation

New Questions in College Statistics