Whenthe standard deviation is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for the mean (u). Method 1: Use the Student's t distribution 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 s as an estimate for standard deviation (o), and then use the standard normal distribution. This method is based on the fact that for large samples, s is a fairly good approximation forthe standard deviation (o). Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution. Consider a random sample of size n = 41, with sample mean x = 44.5 and sample standard deviation s = 4.7. (a) Compute 90%, 95%, and 99% confidence intervals for the mean (u) using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal. 90% 95% 99% lower limit upper limit (b) Compute 90%, 95%, and 99% confidence intervals for the mean (u) using Method 2 with the standard normal distribution. Use s as an estimate for standard deviation (o). Round endpoints to two digits after the decimal. 90% 95% 99% lower limit upper limit

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.5: Comparing Sets Of Data

Problem 13PPS

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Whenthe standard deviation is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for the mean (u).

Method 1: Use the Student's t distribution 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 s as an estimate for standard deviation (o), and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation forthe standard deviation (o). Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 44.5 and sample standard deviation s = 4.7.

(a) Compute 90%, 95%, and 99% confidence intervals for the mean (u) using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

	90%	95%	99%
lower limit
upper limit

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

	90%	95%	99%
lower limit
upper limit

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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