Assume you need to build a confidence interval for a population mean within some

Question

Assume you need to build a confidence interval for a population mean within some given situation. Naturally, you must determine whether you should use either the t-distribution or the z-distribution or possibly even neither based upon the information known/collected in the situation. Thus, based upon the information provided for each situation below, determine which (t-, z- or neither) distribution is appropriate. Then if you can use either a t- or z- distribution, give the associated critical value (critical t- or z- score) from that distribution to reach the given confidence level.

a. 95% confidence n=150 known population data believed to be very skewed Appropriate distribution: Associated critical value: b. 99% confidence n=19 unknown population data believed to be normally distributed Appropriate distribution: Associated critical value: c. 95% confidence n=60 unknown population data believed to be skewed Appropriate distribution: Associated critical value: d. 99% confidence n=12 unknown population data believed to be very skewed Appropriate distribution: Associated critical value:

Explanation / Answer

a)

As sigma is known, and n = 150 is large enough, we use Z DISTRIBUTION. [ANSWER]

Hence,

alpha/2 = (1 - confidence level)/2 =    0.025

Thus, by table/tchnology,

z(alpha/2) =    1.959963985 [ANSWER, CRITICAL VALUE]

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b)

As sigma is unknown, and the sample size is small (n = 19), but the population is normal, then we use T DISTRIBUTION. [ANSWER]

Hence,

df = n - 1 = 18,

alpha/2 = (1 - confidence level)/2 =    0.005

By table/technology,

t(alpha/2) =    2.878440473 [ANSWER, CRITICAL VALUE]

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c)

It depends on your convention in class, but usually, as n = 60, this is large enough, so we can use Z DISTRIBUTION. [ANSWER]

[Please still use the convention you use in class, if it is different.]

Hence,

alpha/2 = (1 - confidence level)/2 =    0.025

Thus, by table/tchnology,

z(alpha/2) =    1.959963985 [ANSWER, CRITICAL VALUE]

****************************

d)

sigma is unknown, and n = 12 is small. We cannot use either distribution. [ANSWER, NEITHER]

[We cannot use either because the central limit theorem cannot be applied yet, as sigma is small and n = 12 is small.]

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