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.