Question

Match each standard deviation with one of the histograms given above.

Answer 1

The question pertains to the analysis of histograms to determine the approximate theoretical distribution of data, with specific emphasis on normal distribution and inference of standard deviation from the histogram's shape.

Step-by-step explanation:

The question revolves around the concepts of sample mean, sample standard deviation, and their comparison to theoretical distributions, particularly in the context of histogram analysis and the assumption of normal distribution. The histogram provides visual representation of the data distribution and, assuming a normal distribution, you would expect a bell-shaped curve centered around the sample mean with spread determined by the standard deviation. A tight, tall histogram suggests a small standard deviation, whereas a wide, short histogram indicates a larger standard deviation.

To compare the sample mean and standard deviation with the theoretical counterparts, you would first calculate these statistics from your dataset. Then you would reflect upon the shape of the histogram to infer the nature of the distribution. If the histogram approximates a bell curve, the distribution of the data can be assumed to be normal, with the sample mean indicative of the theoretical mean and the sample standard deviation reflective of the theoretical standard deviation.

When drawing samples, as stated in the use of n=100 with a known population mean and standard deviation, according to the Central Limit Theorem (CLT), the sample means will distribute normally around the population mean, with a standard deviation equal to the population standard deviation divided by the square root of n (the sample size).

Answer 2

The approximate theoretical distribution can often be assumed normal if the histogram is bell-shaped, and data are collected independently and randomly. The sample mean and standard deviation help understand the center and spread of the data, while the Central Limit Theorem allows us to predict the distribution of sample means.

Step-by-step explanation:

When trying to determine the approximate theoretical distribution using a histogram, we look at the overall shape and spread of the data. Histograms display the frequency of data points within specified ranges and can illustrate the shape of the data distribution.

Sample mean and standard deviation are key metrics in describing that distribution. The sample mean provides a central value around which the data points are spread, while the standard deviation gives an idea of how spread out the data points are from the mean.

If the histogram resembles a bell-shaped curve and the data is collected randomly and independently with a known population mean and standard deviation, we might assume a normal distribution. For instance, if we have a population mean (μ) of 75 and a standard deviation (σ) of 4.5, samples of size n = 100 drawn from this population will distribute normally around the population mean with a standard deviation σ/√n.

Distributions can only be approximated, so we utilize the Central Limit Theorem (CLT) to describe the sampling distribution of the sample means. The CLT states that the distribution of sample means will be approximately normal regardless of the shape of the population distribution, given a sufficiently large sample size (typically n ≥ 30).