Using The Telephone Numbers Listed In Your Local Directory

Using The Telephone Numbers Listed In Your Local Directory A

Using the telephone numbers listed in your local directory as your population, randomly obtain 20 samples of size 3. From each telephone number identified as a source, take the fourth, fifth, and sixth digits. Calculate the mean of the 20 samples. Draw a histogram showing the 20 sample means with classes -0.5 to 0.5, 0.5 to 1.5, 1.5 to 2.5, and so on. Describe the distribution of the sample means in terms of shape, center, and dispersion. Draw 20 additional samples and add the new sample means to the histogram. Describe the evolving distribution and assess its normality using the empirical rule, relating this to the concept of the central limit theorem.

Paper For Above instruction

The exploration of sampling distributions through real data, such as telephone numbers, provides a compelling illustration of fundamental statistical principles, specifically the Central Limit Theorem (CLT). By randomly selecting samples from a population, calculating sample means, and analyzing their distribution, students can observe how sampling distributions tend to approximate a normal distribution, regardless of the original population's distribution, provided the sample size is sufficiently large or numerous samples are taken.

In this exercise, the primary data source is telephone numbers from a local directory. Each participant is instructed to select 20 samples of three telephone numbers randomly, then extract specific digits—namely the fourth, fifth, and sixth digits—from each telephone number. With these digits, the mean of each sample is calculated, resulting in 20 sample means. These means are then visualized via a histogram, categorized into classes ranging from -0.5 to 0.5, 0.5 to 1.5, and so forth. This process simulates the sampling distribution of the mean, allowing for the examination of its shape, central tendency, and variability.

The histogram of the sample means typically exhibits a bell-shaped, symmetric distribution that becomes more pronounced with additional samples. The shape often approximates a normal distribution, aligning with the expectations set by the CLT. The center of this distribution aligns closely with the population mean of the digits, assuming randomness in selected samples, and the dispersion or spread depends on the variability of the underlying digit distribution and the sample size.

Drawing 20 additional samples and adding these new means to the existing histogram allows for observing whether the distribution maintains its normal shape or exhibits skewness or kurtosis. If the distribution remains roughly bell-shaped and symmetric, this further supports the CLT and the notion that the sampling distribution of the mean becomes increasingly normal as sample size or number of samples increases. The empirical rule, which states that approximately 68%, 95%, and 99.7% of data lies within one, two, and three standard deviations from the mean respectively, can be used to assess the normality visually and statistically.

Overall, this activity underscores the importance of understanding sampling distributions, the robustness of the CLT, and how data visualization combined with statistical rules can provide insights into the behavior of sample means derived from real-world data. Such exercises are critical in fostering an intuitive grasp of statistical inference, emphasizing that the distribution of sample means tends toward normality even when the original data may not be normally distributed.

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