Sampling Distributions and Central Limit Theorem Laboratory

Laboratory Activity Overview and Data Structure

• Group Allocations:

  • Students are organized using a random group generator into eight groups (Groups 1 through 8).
  • Group composition consists of one group of two individuals, with all remaining groups consisting of three individuals.

• Variable Definition:

  • The random variable $x$ represents the length of time in days that a recipe lasts at a restaurant before being changed by the chef.
  • The underlying data set consists of $60$ total recipes (numbered 1 through 60).
  • The observations are presented in a table containing four columns of $x$ values.
  • Pre-calculated population parameters are provided directly on the activity sheet for subsequent theoretical comparison:
  • Population mean: $\mu = 3.57$
  • Population standard deviation: $\sigma = 2.17$

Population Data and Initial Histogram Construction

• Histogram Graphical Specifications:

  • Horizontal Axis ($x$-axis): Labeled as "Number of days the recipe lasted."
  • Vertical Axis ($y$-axis): Labeled as "Frequency" (representing the count of recipes), scaled in increments of $5$.
  • Class Boundaries / Bins: Formatted using decimal boundaries offset by $0.5$ to isolate discrete integer values: $0.5, 1.5, 2.5, 3.5, 4.5, 5.5, \dots$
  • Bin Mapping:
  • The interval from $0.5$ to $1.5$ captures recipes that lasted exactly $1$ day.
  • The interval from $1.5$ to $2.5$ captures recipes that lasted exactly $2$ days (excluding $2$ from the preceding bin).
  • Days with zero recorded recipes have a frequency of $0$, represented by a flat line directly on the horizontal axis.
  • For the $1$-day bin ($0.5$ to $1.5$), the counted frequency is $13$ recipes.

• Distribution Shape and Curve Fitting:

  • A smooth continuous curve is traced over the top profile of the histogram bars.
  • The curve exhibits a prominent peak on the lower end with a tail extending toward larger values.
  • The overall shape of the population distribution is identified as skewed to the right (right-skewed).

Sampling Procedure and Central Limit Theorem Simulation ($n = 5$)

• Random Sampling Mechanism:

  • Samples are drawn using a random number generator (e.g., via a mobile search engine or generator tool) configured to produce integers uniformly distributed between $1$ and $60$.
  • Each generated integer identifies the recipe number from the population table, from which the corresponding duration in days ($x$) is recorded.

• Step-by-Step Sample Generation:

  1. Generate five random integers between $1$ and $60$.
  2. Extract the $x$ value (days) for each selected recipe:
  • Example: Recipe #5 lasted $6$ days ($x = 6$).
  • Example: Recipe #48 lasted $11$ days ($x = 11$).
  • Example: Recipe #30 lasted $2$ days ($x = 2$).
  • Example: Recipe #58 lasted $3$ days ($x = 3$).
  • Additional sample pulls observed: Recipe #46, Recipe #38, Recipe #52.
  1. Construct Sample 1 consisting of $n = 5$ observations (e.g., values $11, 2, 5, 2, 3$).
  2. Calculate the sample mean: $\bar{x} = \frac{\sum x}{n}$
  3. Repeat the sampling process independently to produce Sample 2, Sample 3, and Sample 4, resulting in four distinct sample means (e.g., values including $4.8$, $4.4$, etc.).

Group Data Sharing and Sample Mean Distribution Analysis

• Data Pooling:

  • Each group records their four calculated sample means on item number 4 of the activity sheet.
  • Groups pair with the nearest adjacent group to exchange their sets of sample means.
  • Combining both sets produces a collective dataset of $8$ sample means ($N = 8$).

• Statistical Computation of Pooled Means:

  • The grand mean of the eight sample means ($\bar{x}_{\bar{x}}$) is calculated by summing all eight means and dividing by $8$:     $\bar{x}_{\bar{x}} = \frac{\sum \bar{x}}{8}$
  • The sample standard deviation of these eight sample averages ($s_{\bar{x}}$) is determined via spreadsheet software or manual formulas.

• Histogram Construction of Sample Means ($n = 5$):

  • A secondary histogram is established to display the distribution of the pooled sample means.
  • Interval bins are configured in steps of $0.5$, bounded at quarter decimals to avoid ambiguous edge cases:
  • Bin 1: $2.75$ to $3.25$
  • Bin 2: $3.25$ to $3.75$
  • Bin 3: $3.75$ to $4.25$
  • Bin 4: $4.25$ to $4.75$
  • Values falling within intermediate ranges (e.g., $3.3$ and $3.4$) are assigned to the bin spanning $3.25$ to $3.75$.
  • Single occurrences (e.g., a mean of $3.4$) receive a bar height of $1$, while multiple occurrences receive proportional bar heights (e.g., height of $2$ or $3$).

Second Sampling Phase ($n = 10$) and Computational Tools

• Sample Expansion:

  • The sampling protocol is repeated with an increased sample size of $n = 10$ random selections per sample.
  • Sampled recipes and recorded durations include:
  • Recipe #35: $6$ days
  • Recipe #34: $6$ days
  • Recipe #14: $3$ days
  • Recipe #11: $8$ days
  • Recipe #22: $5$ days
  • Recipe #29: $6$ days
  • Recipe #18: $4$ days
  • Recipe #51: $4$ days
  • Recipe #31: $3$ days
  • Recipe #54: $1$ day
  • Recipe #39: $5$ days
  • Recipe #44: $6$ days
  • Recipe #56: $2$ days
  • Additional sampled recipes: #25, #59, #27, #55, #60, #36.
  • Derived sample means for repeated sets yield values such as $3.3$, $3.4$, and $2.4$.

• Spreadsheet Implementation:

  • Mean computation formula: =AVERAGE(data_range)
  • Standard deviation computation formula: =STDEV(data_range)
  • Execution involves highlighting the eight sample mean values and applying =STDEV(...), yielding empirical values such as $0.79$.

Theoretical Sampling Distributions and Formulas

• Central Limit Theorem Parameters:

  • Given a population with mean $\mu$ and standard deviation $\sigma$, the theoretical sampling distribution of the sample mean ($\bar{x}$) for a sample size $n$ possesses:
  • Theoretical Mean of the Sampling Distribution:       $\mu_{\bar{x}} = \mu = 3.57$
  • Theoretical Standard Error of the Sampling Distribution:       $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Theoretical Computations by Sample Size:

  • For Sample Size $n = 5$:$\sigma_{\bar{x}} = \frac{2.17}{\sqrt{5}} \approx 0.9705 \approx 0.97$
  • For Sample Size $n = 10$:$\sigma_{\bar{x}} = \frac{2.17}{\sqrt{10}} \approx 0.6862 \approx 0.69$

• Computational Formatting and Precision:

  • Calculations in handheld calculators require explicit parentheses around the denominator term (e.g., dividing by (\sqrt{5})) to preserve operational order.
  • Standard reporting conventions require rounding numerical results to two decimal places ($2$ decimal digits).

• Distinction Between Sampling Distributions and Normal Probabilities:

  • Defining the sampling distribution entails identifying $\mu_{\bar{x}}$ and calculating $\sigma_{\bar{x}}$.
  • Finding the probability of a sample mean falling within a specified numerical range (e.g., between $85$ and $92$) requires standardizing the interval via $z$-scores:     $z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}}$
  • Standard normal distribution probability tables are not utilized when solely defining the sampling distribution parameters.

Questions & Discussion: Empirical Anomalies versus Theoretical Principles

• Question on Sample Distribution Discrepancies:
  • Prompt: Why does the sample standard deviation for $n = 5$ appear smaller than or close to that of $n = 10$ in the observed student data, and how does this relate to the theoretical distributions and the Law of Large Numbers?
  • Response: Theoretical models dictate that the standard error of the mean must decrease as sample size increases, because the population standard deviation is divided by a larger denominator (since $\sqrt{10} > \sqrt{5}$, $\frac{2.17}{\sqrt{10}} \approx 0.69$ is strictly smaller than $\frac{2.17}{\sqrt{5}} \approx 0.97$).
  • Response on Empirical Data: In small random sample sizes, individual empirical draws may display sampling error where an observed sample standard deviation for $n = 5$ (e.g., $0.79$ or $0.56$) appears smaller than expected. In reporting results, the student must document the exact empirical values obtained from their data while noting that theoretically, an increase in sample size from $n = 5$ to $n = 10$ systematically reduces the spread of the sampling distribution due to division by a larger root divisor.