 # Confidence Intervals: Estimating Population Mean

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Confidence Intervals: Estimating Population Mean
Section 8.3 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Lesson 6.1.1

Objectives Confidence Intervals: when σ is known: The One-Sample Z-Interval for a Population Mean Confidence Intervals: when σ is Unknown: The t- Distribution Calculator Conditions for Calculating Confidence Intervals Following 4 Step Process Robust

So… Inference about a population proportion usually arises when we study categorical variables. We learned how to construct and interpret confidence intervals for an unknown parameter p in Section 8.2. To estimate a population mean, we have to record values of a quantitative variable for a sample of individuals. It makes sense to try to estimate the mean amount of sleep that students at a large high school got last night but not their mean eye color! In this section, we’ll examine confidence intervals for a population mean μ.

One Sample Z Interval for a Population Mean: When σ is Known
Draw a SRS of size n from a population having unknown mean µ and known standard deviation σ. As long as the Normal and Independent conditions are met, a level C confidence interval for µ is: 𝑥 ± 𝑧 ∗ σ 𝑛 The critical value 𝑧 ∗ is found from the standard Normal distribution.

Choosing the Sample Size
To determine the sample size n that will yield a level C confidence interval for a population mean with a specified margin of error ME, solve the following inequality for n: 𝑧 ∗ σ 𝑛 ≤ ME Lets look at an example from a hand out

To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean (this mean is 10 grams if the scale has no bias). In previous studies, the standard deviation of the scale readings has been about gram. How many measurements must be averaged to get a margin of error of with 98% confidence? Show your work. Solution:

When σ is unknown: The t-distribution
When we knew all of the information to find a z score with our formula of: 𝑧= 𝑥 −µ σ 𝑛 But what if we don’t know the standard deviation anymore? ??= 𝑥 −µ ?? 𝑛 When we don’t know σ, we estimate it using the sample standard deviation 𝑠 𝑥 . This is known as t-score of a t-distribution t= 𝑥 −µ 𝑠 𝑥 𝑛

The t-distribution for a Population Mean: When σ is Unknown
Draw a SRS of size n from a large population that has a Normal distribution with mean µ and standard deviation σ. t= 𝑥 −µ 𝑠 𝑥 / 𝑛 Has the t distribution with degrees of freedom, df=n-1. This statistic will have approximately a normal distributions as long as the sampling distribution of 𝑥 is close to Normal.

Degrees of Freedom The figure below compares the density curves of the standard Normal distribution and the t distributions with 2 and 9 degrees of freedom. The figure illustrates these facts about the t distributions:

Degree of Freedom Density curves for the t distributions with 2 and 9 degrees of freedom and the standard Normal distribution. All are symmetric with center 0. The t distributions are somewhat more spread out. The density curves of the t distributions are similar in shape to the standard Normal curve. They are symmetric about 0, single-peaked, and bell-shaped. The spread of the t distributions is a bit greater than that of the standard Normal distribution. The t distributions in the figure have more probability in the tails and less in the center than does the standard Normal. This is true because substituting the estimate sx for the fixed parameter σ introduces more variation into the statistic. As the degrees of freedom increase, the t density curve approaches the standard Normal curve ever more closely. This happens because sx estimates σ more accurately as the sample size increases. So using sx in place of σ causes little extra variation when the sample is large.

Finding 𝑡 ∗ Problem: Suppose you want to construct a 95% confidence interval for the mean μ of a Normal population based on an SRS of size n = 12. What critical value t* should you use? Solution: In Table B, we consult the row corresponding to df = n − 1 = 11. We move across that row to the entry that is directly above 95% confidence level on the bottom of the chart. The desired critical value is t* =

Calculator in Finding 𝑡 ∗
Most newer TI-84 and TI-89 calculators allow you to find critical values t* using the inverse t command. As with the calculator’s inverse Normal command, you have to enter the area to the left of the desired critical value. TI 84: 2nd VARS (DIST) > 4:invT( > complete the command invT(.975,11) > Enter TI 89: In Statistics/List Editor > F5> 2:Inverse>2:Inverse t… in the dialog box, enter Area: .975 and Deg of Freedom df : 11> Enter

Check For Understanding
Use Table B to find the critical value t* that you would use for a confidence interval for a population mean μ in each of the following situations. If possible, check your answer with technology. (a) A 98% confidence interval based on n = 22 observations. Correct Answer ….t* = 2.518 (b) A 90% confidence interval from an SRS of 10 observations. Correct Answer…. t* = 1.833 (c) A 95% confidence interval from a sample of size 7. Correct Answer….t* = 2.447

AP Tip On the AP exam, if the desired degree of freedom is not included in the table, I is acceptable to use the next smaller df in the table or technology.

The One Sample t Interval for a Population Mean
Choose an SRS of size n from a population having unknown mean μ. A level C confidence interval for μ is statistic ± (critical value) · (standard deviation of statistic) where t* is the critical value for the tn−1 distribution. Use this interval only when (1) the population distribution is Normal or the sample size is large (n ≥ 30), and (2) the population is at least 10 times as large as the sample.

Standard Error The Standard error of the sample mean 𝑥 is 𝑠 𝑥 𝑛 , where 𝑠 𝑥 is the sample standard deviation. It describes how far 𝑥 will be from µ, on average, in repeating SRSs of size n. Standard Error = 𝑠 𝑥 𝑛

Calculator: One sample t interval
STAT> TESTS> 8:Tinterval… from here adjust your settings to your problem > Enter TI 89: In Statistics/List Editor > 2nd F2 (F7) > interval (ints) >2:Tinterval… Choose “data” as the data input method> adjust your settings as needed> Enter

Conditions Random: The data come from a random sample of size n from the population of interest or a randomized experiment. This condition is very important. Normal: The population has a Normal distribution or the sample size is large (n ≥ 30). Independent: The method for calculating a confidence interval assumes that individual observations are independent. To keep the calculations reasonably accurate when we sample without replacement from a finite population, we should check the 10% condition: verify that the sample size is no more than 1/10 of the population size.

AP Exam Common Error When students are constructing a one-sample t interval for a population mean based on a small sample, many students neglect to graph the sample data and to comment on the Normal conditions. Simply saying that we “assume normality” will not earn full credit when the data are provided. Students must show a graph and give an appropriate comment that addresses the normality of the population

Remember to follow the A Four-Step Process
State: what parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, preform calculations. Conclude: Interpret your interval in the context of the problem. AP expects ALL FOUR, do not skip step 1!

Robust The stated confidence level of a one-sample t interval for μ is exactly correct when the population distribution is exactly Normal. No population of real data is exactly Normal. The usefulness of the t procedures in practice therefore depends on how strongly they are affected by lack of Normality. An inference procedure is called robust when the procedures are violated, yet the results are still fairly accurate.

Objectives Confidence Intervals: when σ is known: The One-Sample Z-Interval for a Population Mean Confidence Intervals: when σ is Unknown: The t- Distribution Calculator Conditions for Calculating Confidence Intervals Following 4 Step Process Robust

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