Confidence Intervals for Population Mean (Gas Prices in Orange County)

Notation and Key Concepts

  • Population parameter for numerical variables: the population mean, denoted by \mu.
  • Population standard deviation is denoted by \sigma.
  • Sample statistic used to estimate the population mean: the sample mean, denoted by \bar{x}.
  • Sample standard deviation is denoted by s.
  • Sample size is denoted by n.
  • When you have data in a dataset, you can obtain the key values from the dataset summary: the sample mean \bar{x}, the sample standard deviation s, and the sample size n.

Confidence Interval Basics

  • A confidence interval is a set of values that could plausibly contain the population parameter. In this context, it is the set of possible values for the population mean \mu.
  • We typically choose a confidence level, which represents how confident we want to be about our interval capturing the true parameter. Common levels are 90%, 95%, 98%, and 99%; in this class the focus is on 95%.
  • The general form of a confidence interval for the mean (with the 95% approach shown in the transcript) is:
    \text{CI} = [\bar{x} - \text{ME}, \; \bar{x} + \text{ME}]
  • The margin of error (ME) used in the example is given by:
    \text{ME} = 2 \cdot \frac{s}{\sqrt{n}}
  • The standard error (the standard deviation of the sampling distribution of the mean) is:
    \text{SE} = \frac{s}{\sqrt{n}}
  • The two-sided confidence interval reflects a range of plausible values for the population mean based on the sample data.
  • Interpretation (as described in the transcript):
    • We are 95% confident that the true population mean lies between the lower and upper bounds of the interval. In other words, if we repeated the study many times and built a 95% CI each time, about 95% of those intervals would contain the true mean.
    • The interval is the set of plausible values for the population mean given the observed sample.

Worked Example: Orange County gas stations

  • Given sample data (from the transcript):
    • Sample size: n = 25
    • Sample mean: \bar{x} = 4.32
    • Sample standard deviation: s = 0.19
  • Calculate the standard error and margin of error:
    • Standard error:
      \text{SE} = \frac{s}{\sqrt{n}} = \frac{0.19}{\sqrt{25}} = \frac{0.19}{5} = 0.038
    • Margin of error (using the 95% rule with multiplier 2):
      \text{ME} = 2 \cdot \text{SE} = 2 \cdot 0.038 = 0.076
  • Construct the confidence interval:
    • Lower bound:
      \bar{x} - \text{ME} = 4.32 - 0.076 = 4.244 \approx 4.24
    • Upper bound:
      \bar{x} + \text{ME} = 4.32 + 0.076 = 4.396 \approx 4.40
    • Confidence interval (95%):
      \left[4.24, \; 4.40\right]
  • Interpretation: We are 95% confident that the average gas price of all Orange County gas stations lies between 4.24 and 4.40.
  • Notes on rounding: The margin of error computed as 0.076 rounds to two decimals as 0.08, which would yield an interval of roughly [4.24, 4.40] when applied to the rounded mean, consistent with the transcript.

Step-by-Step Procedure to Construct the CI (summary from the lecture)

1) Compute the standard error of the mean (sampling mean):
\text{SE} = \frac{s}{\sqrt{n}}\,.
2) Compute the margin of error: (using the 95% rule from the lecture)
\text{ME} = 2 \cdot \text{SE} = 2 \cdot \frac{s}{\sqrt{n}}\,.
3) Compute the interval limits:

  • Lower limit: \bar{x} - \text{ME}
  • Upper limit: \bar{x} + \text{ME}
    4) State the interpretation: We are 95% confident that the true population mean lies between the lower and upper bounds.

Using the Dataset Summary to Gather Inputs

  • To obtain \bar{x}, s, and n, right-click the dataset and choose Dataset Summary. The summary shows:
    • Number of observations (which gives n)
    • The sample mean \bar{x}
    • The sample standard deviation s
  • Example from the transcript: 25 gas prices, \bar{x} = 4.32, s = 0.19, n = 25.

Practical Notes and Context

  • The 95% confidence level is the most commonly used in the session; higher levels (e.g., 98%, 99%) would make the interval wider, and lower levels (e.g., 90%) would make it narrower.
  • The lecture emphasizes that the calculation uses the number 2 as the multiplier for ME (an approximate z-value for 95% in large samples). In practice, with known or assumed population SD, you might use the exact critical value (e.g., 1.96 for 95%), or a t-value when using the sample SD and small samples. Here, the taught approach uses 2 as a convenient default.
  • The interpretation includes that any value within the interval is plausible as an estimate for the true mean, but the true mean is a fixed value; the randomness is in the sampling process and the interval produced from a given sample.
  • The material also touches on comparing intervals across groups (as seen in the discussion of question 14 in the worksheet), which is a way to assess whether groups have similar or different mean levels.
  • In a classroom workflow, process steps are reinforced: compute SE, then ME, then lower/upper limits, then interpret the interval.