Which, If Either, Is Larger, The Cumulative Distribution Of

Which, if either, is larger, the cumulative distribution of T(8) at -3.2 or the cumulative distribution of T(27) at -3.2

The problem requires us to compare the values of the cumulative distribution function (CDF) of two Student's t-distributions, specifically T(8) and T(27), at the point -3.2. The Student's t-distribution is symmetric about zero, with a shape that depends on its degrees of freedom (df). For a given value x

When examining the t-distribution at a negative value such as -3.2, a lower degree of freedom (df) results in a distribution with heavier tails. This means that for the same negative value, the CDF of T(8) would be larger than that of T(27), because the heavier tails increase the probability of observing values further from the mean. Conversely, the distribution with higher degrees of freedom, such as T(27), is more concentrated around the mean, making the probability of observing a very negative value like -3.2 smaller.

Hence, at -3.2, the cumulative distribution of T(8) exceeds that of T(27), implying that the probability of obtaining a value less than -3.2 is higher for T(8) than for T(27). This conclusion aligns with the properties of the t-distribution: lighter tails as df increases lead to lower probabilities in the tails. Therefore, the answer is that the cumulative distribution of T(8) at -3.2 is larger than that of T(27) at -3.2.

Which, if either, is larger, the cumulative distribution of N(0,1) at -3.2 or the cumulative distribution of T(8) at -3.2

Comparing the standard normal distribution N(0,1) with the Student's t-distribution T(8) at -3.2 involves understanding how their tail probabilities differ. The standard normal distribution is characterized by lighter tails compared to the t-distribution, which has heavier tails for lower degrees of freedom, such as 8. For the same negative value -3.2, the probability in the tails of the distribution captures how likely it is to observe such an extreme value or less.

Because the t-distribution with df=8 has heavier tails than the standard normal distribution, the probability of observing a value less than -3.2 (i.e., the CDF at -3.2) for T(8) is greater than that for N(0,1). This explains why, at the same negative point, the cumulative distribution function of T(8) will be larger than that of the standard normal distribution.

Concretely, the heavier tails mean more mass in the extremes; thus, the cumulative probability up to -3.2 for T(8) exceeds that of the standard normal. Consequently, the cumulative distribution of N(0,1) at -3.2 is smaller than that of T(8) at -3.2.

Inference about Variance and Hypothesis Testing

When aiming to reject the null hypothesis (H0) in a hypothesis test, it is advantageous for the estimator of variance to be small. A smaller estimated variance implies that the sample mean is more precise, with less variability, leading to a narrower confidence interval and a higher likelihood of detecting a true effect if one exists. This increases the power of the test because the test statistic becomes more sensitive to deviations from H0, allowing for easier rejection when the alternative hypothesis is true.

In statistical testing, the test statistic often involves the estimated variance in the denominator; hence, a smaller variance amplifies the test statistic's magnitude for a given effect size, increasing the chance of surpassing the critical value needed to reject H0. Conversely, a large estimated variance introduces more uncertainty, making it harder to achieve statistically significant results. Therefore, an estimate of variance that is small enhances the capacity to reject H0 when appropriate.

Estimating the Population Mean Time with a Confidence Interval

Given a sample of 60 observations of a laboratory procedure's duration, with a mean of 20.32 minutes and a standard deviation of 3.82 minutes, we aim to construct a 95% confidence interval for the true mean time. First, we identify the standard error (SE) of the mean as:

SE = s / √n = 3.82 / √60 ≈ 0.492

Using the t-distribution, since the population standard deviation is unknown and the sample size is moderate (n=60), the degrees of freedom are 59. The critical t-value for a 95% confidence level and df=59 is approximately 2.00 (from t-tables or software).

Thus, the margin of error (ME) is:

ME = t0.975, 59 × SE ≈ 2.00 × 0.492 ≈ 0.984

The 95% confidence interval for the population mean is therefore:

20.32 ± 0.984, which is (19.336 minutes, 21.304 minutes)

This interval indicates that there is a 95% probability that the true mean time to complete the procedure falls within this range. The interval's width reflects the precision of the estimate — the smaller the standard deviation or the larger the sample size, the narrower the interval, signifying greater confidence in the estimate.

In practical terms, this means that if the same study were repeated numerous times, approximately 95% of the calculated confidence intervals would contain the true average time required for the procedure. This quantification of uncertainty is fundamental in statistical inference when estimating population parameters based on sample data.

References

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  • Ghasemi, A., & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide for Non-Statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486-489.
  • Higgins, J. P. T., & Green, S. (2011). Cochrane Handbook for Systematic Reviews of Interventions. Cochrane Collaboration.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
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  • Student. (1908). The Probable Error of a Mean. Biometrika, 6(1), 1–25.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.

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