When ? is unknown and the sample is of size n ?30, there are two methods for com

Question

When ? is unknown and the sample is of size n?30, there are two methods for computing confidence intervals for ?.

Method 1: Use the Student's tdistribution with d.f.= n?1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n?30, use the sample standard deviation sas an estimate for ?, and then use the standard normal distribution.
This method is based on the fact that for large samples, sis a fairly good approximation for ?. Also, for large n, the critical values for the Student's tdistribution approach those of the standard normal distribution.

Consider a random sample of size n= 31, with sample mean x= 44.2and sample standard deviation s = 6.2.

(a) Compute 90%, 95%, and 99% confidence intervals for ?using Method 1 with a Student's tdistribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for ?using Method 2 with the standard normal distribution. Use sas an estimate for ?. Round endpoints to two digits after the decimal.

(c) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for ?using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(e) Compute 90%, 95%, and 99% confidence intervals for ? using Method 2 with the standard normal distribution. Use sas an estimate for ?. Round endpoints to two digits after the decimal.

Explanation / Answer

Solution ?? µ?2

Let X be the random variable representing the characteristic under the question.

Mean X = µ SD(X) = ?

Sample size = n, Sample mean = Xbar, Sample SD = s.

Back-up Theory

100(1 - ?) % Confidence Interval for ?, when ? is not known is:

i) Small sample: Xbar ± (tn- 1, ? /2)s/?n where tn – 1, ? /2 = upper (? /2)% point of

t-distribution with (n - 1) degrees of freedom……………………………………… (1)

ii) Large sample:Xbar ± (Z? /2)s/?n   where Z? /2 = upper (? /2)% point of N(0, 1). …(2)

Now, to work out the answer,

Part (a)

90%, 95%, and 99% confidence intervals for using Method 1 with a Student's t-distribution.

Given n = 31, xbar = 44.2, s = 6.2, t30, 0.05 = 1.697, t30, 0.025 = 2.042, t30, 0.005 = 2.750 [all from standard t-distribution tables.] So, vide (1) under Back-up Theory,

90% CI is: [42.31, 46.09] ANSWER 1

95% CI is: [41.93, 46.47] ANSWER 2

99% CI is: [41.14, 47.26] ANSWER 3

Part (b)

90%, 95%, and 99% confidence intervals for ? using Method 2 with the standard normal distribution.

Given n = 31, xbar = 44.2, s = 6.2, Z 0.05 = 1.645, Z 0.025 = 1.96, Z 0.005 = 2.575 [all from standard Normal tables.] So, vide (2) under Back-up Theory,

90% CI is: [42.37, 46.03] ANSWER 1

95% CI is: [42.02, 46.38] ANSWER 2

99% CI is: [41.33, 47.07] ANSWER 3

DONE

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