Textbook Question
True or False: To cut the standard error of the mean in half, the sample size must be doubled.
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7. Sampling Distributions & Confidence Intervals: Mean
Sampling Distribution of the Sample Mean and Central Limit Theorem
1:42 minutesProblem 9.2.4
Textbook Question
Put the following in order from least to greatest.
- t0.10 with 5 degrees of freedom
- t0.10 with 15 degrees of freedom
- z0.10
Verified step by step guidance1
Understand that the notation t0.10 with a certain degrees of freedom (df) refers to the t-distribution critical value where the area to the right of the value is 0.10. Similarly, z0.10 refers to the z-distribution (standard normal) critical value with 0.10 area to the right.
Recall that the t-distribution is symmetric and bell-shaped like the normal distribution but has heavier tails, especially with fewer degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
Identify that for the same right-tail probability (0.10), the critical value t0.10 with fewer degrees of freedom will be larger in magnitude than with more degrees of freedom, because heavier tails mean more extreme values are needed to capture the same tail area.
Find or look up the critical values for each distribution: t0.10 with 5 df, t0.10 with 15 df, and z0.10. These values are all positive numbers representing the cutoff points where the area to the right is 0.10.
Compare the three critical values numerically and then order them from least to greatest based on their magnitude.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
t-Distribution and Degrees of Freedom
The t-distribution is a probability distribution used when the sample size is small and the population standard deviation is unknown. Its shape depends on degrees of freedom (df), with fewer df resulting in heavier tails. As df increases, the t-distribution approaches the standard normal distribution.
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Critical Values and Tail Probabilities
A critical value corresponds to a point on a distribution where the tail area equals a specified probability (e.g., 0.10). For t and z distributions, t0.10 or z0.10 represents the value where 10% of the distribution lies to the right, used in hypothesis testing and confidence intervals.
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Comparison of t and z Critical Values
For the same tail probability, t-distribution critical values are larger than z-values when degrees of freedom are low, reflecting greater uncertainty. As degrees of freedom increase, t critical values decrease and approach the z critical value, allowing ordering from least to greatest based on df.
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