Question

Suppose the preliteracy scores of three-year-old students in the United States are normally distributed. Shelia, a preschool teacher, wants to estimate the mean score on preliteracy tests for the population of three-year-olds. She draws a simple random sample of 20 students from her class of three-year-olds and records their preliteracy scores (in points).

80,81,89,90,91,91,91,94,95,95,95,101,101,102,102,104,104,108,109,11180,81,89,90,91,91,91,94,95,95,95,101,101,102,102,104,104,108,109,111

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Calculate the sample mean (?⎯⎯⎯x¯), sample standard deviation (?s), and standard error (SE) of the students' scores. Round your answers to four decimal places.

Determine the ?t-critical value (?t) and margin of error (?m) for a 90% confidence interval. Round your answers to three decimal places.

What are the lower and upper limits of a 90% confidence interval? Round your answers to three decimal places.

?⎯⎯⎯=x¯=

?=s=

?t =

SE=SE=

?m =

lower limit:

upper limit:

Which is the correct interpretation of the confidence interval?

There is a 90% chance that the population mean is between 93.350 points and 100.050 points.

Shelia is 90% confident that the true population mean is between 93.350 points and 100.050 points.

Shelia is 90% confident that the true population mean is between 95.513 points and 99.887 points.

Shelia is certain that the true population mean is between 93.350 points and 100.050 points.

There is a 90% chance that the true population mean is between 95.513 points and 99.887 points.

Answer 1

Solution:

Sample size = n = 20

Part a) Calculate the sample mean , sample standard deviation ( s), and standard error (SE)

Formula:

Thus we need to make following table:

x	x^2
80	6400
81	6561
89	7921
90	8100
91	8281
91	8281
91	8281
94	8836
95	9025
95	9025
95	9025
101	10201
101	10201
102	10404
102	10404
104	10816
104	10816
108	11664
109	11881
111	12321

Thus

and

Part b) Determine the t-critical value (t) and margin of error ( m) for a 90% confidence interval.

df   = n - 1 = 20 - 1 = 19

Two tail area = 1 - 0.90 = 0.10

t critical value = 1.729

Margin of Error is:

Part c) What are the lower and upper limits of a 90% confidence interval?

and

Part d) Which is the correct interpretation of the confidence interval?

Shelia is 90% confident that the true population mean is between 93.350 points and 100.050 points.