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Error and Confidence Solutions

Note

Inferential statistics generalize, or infer, from a sample what is likely to happen in the population. See Lesson 47 to review.

• In inferential statistics, a sample is used because    collecting data from an entire population    (a census) is impractical.

• However, using a sample results in a    level of variation    (or uncertainty) in the estimates. 

• The variability between a random sample and a population produces a    sampling error   .

• A key principle in statistics is that    the larger the sample, the smaller the sampling error   , because a larger sample more accurately represents the population.

Maximum Error of the Estimate (Margin of Error)

• The margin of error (or maximum error of the estimate) is the estimated difference between a    point estimate    (sample mean or sample proportion) and the    true population value   .
• A point estimate is a specific numerical value estimate of a    population parameter   .
• The formula for the maximum error of the estimate, E, is: $E=z·\frac{s}{\sqrt{n}}$

  • z:    the confidence level as a z-score
   
  • s:    sample standard deviation (or population standard deviation)   
  • n: $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{sample}\mathbf{ }\mathbf{size}\mathbf{,}\mathbf{ }\mathit{n}\mathbf{\ge }\mathbf{30}\mathbf{ }\mathbf{ }\mathbf{ }}$

• The maximum error of the estimate (or margin of error) is a single value that is used alongside a sample statistic to    calculate confidence   .
• The confidence in the reliability of the sample to estimate the population parameter can be expressed in two ways:

  •    Confidence level   
  •    Confidence interval   

Note

In this lesson, the focus is the confidence in estimating the population mean, μ.

Confidence Level

• The confidence level is the probability that the estimate    reliably includes the true parameters   .
• Confidence levels are written as percentages:

• As the confidence level    increases   , the precision of estimating the true population mean    decreases   .

Confidence Interval

• A confidence interval, CI, is the    range of values    that represent the    estimate of a population parameter   .
• It accounts for sampling error, because a CI is written as a    range of values    instead of a single point estimate.

• Therefore, population mean, μ,    is inferred    from the confidence interval.

• It is calculated with a sample mean, $\overline{x}$, and the    maximum error of the estimate, E   .
• Once the maximum error of the estimate is calculated, the confidence interval is written as a    compound inequality   : $\overline{x}–E \le \mu \le \overline{x}+E$ or $\overline{x}\pm E$
  • $\overline{x}$:    “x-bar,” the sample mean   
  • μ:    population mean   
  • E:    the maximum error of the estimate   
• Remember, the    larger    the interval, the more confident you can be that the population mean will be found within it.

Example 1

In 2023, a random sample of 60 retail employees in a large city found the sample mean wage was $17.25 per hour with a standard deviation of $3.50 per hour. Calculate the confidence intervals (CI) with 90% and 99% certainty for the population mean. Explain.

$\begin{array}{lll}s=3.50,& n=60& \\ \mathbf{90}\mathbf{\%}\mathbf{ }\mathbf{CI}& & \mathbf{99}\mathbf{\%}\mathbf{ }\mathbf{CI}\\ z=1.645& & z=2.576\\ E=0.75& & E=1.16\end{array}$

Note

The CI at 90% is more precise, and the CI at 99% is more inclusive.

Rounding to an additional decimal place does not make sense for this problem because it pertains to money.

Example 2

At the Zander Zoo, a random survey of 350 visitors found that 83.7% are satisfied with the maps for navigating the zoo. Find the interval that most likely contains the population parameter for visitors dissatisfied with the zoo’s maps for a margin of error of $\pm 5.34\%$.

$$\begin{gathered} \begin{array}{l}100–83.7=16.3\% \\ 16.3–5.34=10.96\%\\ 16.3+5.34=21.64\%\end{array} \end{gathered}$$

It is likely that    between 10.96% and 21.64%    of the zoo’s visitors are dissatisfied with the maps.

Example 3

A national movie theater chain wants to estimate the average number of ounces of popcorn purchased per movie ticket sale. From years of data, they know the population standard deviation for popcorn purchases per ticket sale is 4.5 ounces. The chain wants to be 95% confident in their estimate and decides to take a random sample of 100 movie ticket sales.

Note

Remember that statistics round to one place value more than the given information.

Example 4

A Bright Idea manufactures light bulbs and needs to estimate the average lifespan of a new type of bulb. They know from pilot studies that the standard deviation (s) of a light bulb’s lifespan is 120 hours. They want to be 99% confident that their estimate is within 25 hours of the true average lifespan. What is the minimum sample size that can be tested to reach a 99% confidence level for A Bright Idea?