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Explain why the​ t-distribution has less spread as the number of degrees of freedom increases. Choose the correct answer below. A. The​ t-distribution has less spread as the degrees of freedom increase because the variability introduced into the​ t-statistic becomes greater as n increases. B. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to by the law of large numbers. C. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, less information is known about by the law of large numbers. D. The​ t-distribution has less spread as the degrees of freedom increase​ because, for large values of​ n, n​30, the​ t-distribution and the normal distribution are the same.

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Answer:

B. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to sigma by the law of large numbers.

Step-by-step explanation:

The sample standard deviation, s gets considerably closer to the population standard deviation, σ, this trend follows the proposition of the law of large numbers whereby the mean or average value changes as the sample size, n increases. According to the law of large numbers, as the sample size increases, the sample mean gets continously closer to the population mean, sample standard deviation follows this same trend and thus variability or spread decreases as sample size increases.

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The plausible reason that explains why the t-distribution will have a less spread as the number of degrees of freedom increases is because: as n​ increases, s becomes closer to by the law of large numbers. (Option B)

Note the following:

  • Based on the theorem of large number, the population mean and sample mean will get closer as the sample size increases.
  • By implication, the sample standard deviation, s, and the population standard deviation, σ becomes closer too, thus, the spread will reduce as the sample size increases.

Therefore, the plausible reason that explains why the t-distribution will have a less spread as the number of degrees of freedom increases is because: as n​ increases, s becomes closer to by the law of large numbers. (Option B)

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