Confidence Intervals (A level)Firstly let me apologise for asking this question on here - I see this as getting a sledgehammer to crack a nut, but I have nobody else that I can ask for advice on this topic.I'm an A level Mathematics student so please forgive me for lack of/poor notation that you may normally come to expect/be familiar with.I am looking at the specification for my exam that is coming up and these are three sections:A) Construct symmetric confidence intervals for the mean of a normal distribution with known varianceB) Construct symmetric confidence intervals from large samples, for the mean of a normal distribution with unknown variance.C) Construct symmetric confidence intervals from small samples, for the mean of a normal distribution with unknown variance using the t -distribution.

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Confidence Intervals (A level)Firstly let me apologise for asking this question on here - I see this as getting a sledgehammer to crack a nut, but I have nobody else that I can ask for advice on this topic.I&#039;m an A level Mathematics student so please forgive me for lack of/poor notation that you may normally come to expect/be familiar with.I am looking at the specification for my exam that is coming up and these are three sections:A) Construct symmetric confidence intervals for the mean of a normal distribution with known varianceB) Construct symmetric confidence intervals from large samples, for the mean of a normal distribution with unknown variance.C) Construct symmetric confidence intervals from small samples, for the mean of a normal distribution with unknown variance using the t -distribution.

Marvin Hughes · Accepted Answer

Step 1The difference between B and C is in the choice of critical value. A typical symmetric confidence interval for a location parameter has the form  point estimate  ±  critical value  ×  standard error  ,where in turn  standard error  =      standard deviation          sample size        ,and  critical value  ×  standard error  =  margin of error  .The choice of critical value is informed by the nature of the sampling distribution. When a normally distributed population has unknown variance, the sampling distribution of the mean is Student&#039;s t-distributed; however, when the sample size is large, the difference in critical values is negligible; i.e.,      lim          ν      →      ∞            t          ν      ,      α        ∗    =      z          α        ∗    ,where       t          ν      ,      α        ∗   is the upper   α quantile of the Student&#039;s t distribution with   ν degrees of freedom, and       z    α    ∗   is the upper   α quantile of the standard normal distribution. So for scenario B, you will use       z          α              /            2        ∗   for a two-sided CI with large sample size as an approximation; for scenario C, you will use       t          ν      ,      α              /            2        ∗   for the critical value.Step 2The difference between A and B lies in the standard deviation. In scenario A, it is presumed to be known, thus the standard error will use the population standard deviation   σ in the numerator. In scenario B, it is unknown, thus you will use the unbiased estimator of the standard deviation  s  =            1              n        −        1                    ∑              i        =        1            n        (          x      i        −                  x        ¯                    )      2        .In practice, the factor of   n  −  1 rather than n makes little difference when n is large as in this case. But in scenario C, as   σ is also unknown and must be estimated from the sample, you must use   n  −  1, otherwise your CI will not have the required coverage probability even if you correctly use the t-distribution quantile for the critical value.

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