Construct the confidence interval for the population mean muμ.

Construct the confidence interval for the population mean mu. c = 0.90, x= 9.5, s = 0.6, and n = 51 A 90% confidence interval for mu is . Round to two decimal place as need.

Answer

General guidance

Concepts and reason
Confidence interval: A range of values such that the population parameter can expected to contain for the given confidence level is termed as the confidence interval. In other words, it can be defined as an interval estimate of the population parameter which is calculated for the given data based on a point estimate and for the given confidence level. Moreover, the confidence level indicates the possibility that the confidence interval can contain the population parameter. Usually, the confidence level is denoted by . The value is chosen by the researcher. Some of the most common confidence levels are 90%, 95%, and 99%. For testing the mean, If the sample size is large and population standard deviation is known, then the Z test is used. Normal distribution: The Normal distribution has a bell shaped curve and it is a continuous distribution. The normally distributed random variable x has mean and standard deviation . Margin of Error: The margin of error is defined as a statistic which gives the amount of sampling error in the given study. Also, the margin of error tells the percentage of points that the obtained results would differ from that of the given population value.

Fundamentals

The confidence interval for Mean (one sample z test): confidence interval = xza Here, X: Sample mean
za critical value
0: Population standard deviation
n: Sample size The formula of sample size is, Where, is the population standard deviation, E is the margin of error and is the critical value at level of significance. The critical value is obtained by ‘Standard normal table’. Procedure for finding the Z-value is listed below: 1.From the table of standard normal distribution, locate the probability value. 2.Move left until the first column is reached. 3.Move upward until the top row is reached. 4.Locate the probability value, by the intersection of the row and column values gives the area to the left of z.

Step-by-step

Step 1 of 2

The critical value for Za=0.10) is obtained below: Here, confidence level is 0.90. For(1-a)=0.90
a=0.10
= 0.05
1- = 0.95 From the Standard Normal Table, the required Za=0.10) The value Za=0.10) is obtained below: That is, P(Z = z)=0.95 Procedure for finding the z-value is listed below: 1.From the table of standard normal distribution, locate the probability value of 0.95. 2.Move left until the first column is reached. Note the value as 1.6. 3.Move upward until the top row is reached. Note the value as 0.045. 4.The intersection of the row and column values gives the area to the two tail of z. That is, P(Z 51.645)=0.95 From standard normal table, the required Za=0.10) value for 95% confidence level is 1.645.

Locate the probability of 0.95 in the standard normal table and identify the corresponding row and column values to obtain the value ofZa=0.10).

Step 2 of 2

The 90% confidence interval for mu is obtained below: From the given information, the confidence interval is 90%, and the value are n=51,7 = 9.5, and s = 0.6. The sample size is, confidence interval =9.5+(
=9.5+(1.645x0.0840)
=9.5+0.14
=(9.5-0.14,9.5+0.14)

= (9.36.9.64)

The 90% confidence interval for the mu is (9.36.9.64).


The 90% confidence interval for the mu is obtained by substituting the corresponding values of , , n and in the confidence interval formula.

Answer

The 90% confidence interval for the mu is (9.36.9.64).

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