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Chapter 8: Problem 26

Determine the \(t\) -percentile that is required to construct each of the following one-sided confidence intervals: (a) Confidence level \(=95 \%,\) degrees of freedom \(=14\) (b) Confidence level \(=99 \%,\) degrees of freedom \(=19\) (c) Confidence level \(=99.9 \%,\) degrees of freedom \(=24\)

Short Answer

Expert verified
(a) t = 1.761, (b) t = 2.539, (c) t = 3.745.

Step by step solution

01

Understanding the t-percentile

The t-percentile is a value that can be found on the t-distribution table, which is used to determine the critical value for constructing confidence intervals based on sampled data. The t-distribution depends on two parameters: the confidence level and the degrees of freedom.
02

Calculate the Complementary Alpha Level

For a one-sided confidence interval, subtract the confidence level from 100% to find the complementary alpha level, 伪. This gives: (a) 伪 = 100% - 95% = 5% or 0.05 (b) 伪 = 100% - 99% = 1% or 0.01 (c) 伪 = 100% - 99.9% = 0.1% or 0.001.
03

Determine the t-Percentile for Each Case

Using the alpha level determined in Step 2 and the provided degrees of freedom, consult the t-distribution table to find the t-percentile (critical value). - For (a) with 伪 = 0.05 and df = 14, find the t-value for the 95th percentile. (b) with 伪 = 0.01 and df = 19, find the t-value for the 99th percentile. (c) with 伪 = 0.001 and df = 24, find the t-value for the 99.9th percentile. - Use standard t-distribution tables which provide these percentile values based on degrees of freedom.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Intervals
Confidence intervals offer a way to express how well we expect sampled data to estimate a true population parameter. They create a range around a sample statistic within which we expect the true population parameter to fall, with a certain level of confidence (e.g., 95%, 99%). For example, if we create a 95% confidence interval, we are saying that if we were to draw 100 different samples and build a confidence interval from each, about 95 of those intervals would contain the true population parameter. Key points about confidence intervals:
  • They are affected by sample size: Larger samples tend to produce more precise (narrower) intervals.
  • The confidence level represents the probability that the interval produced will contain the true parameter.
  • Confidence intervals use the t-distribution when the population standard deviation is unknown, and the sample size is small.
Confidence intervals are crucial when making inferences about populations from samples.
Degrees of Freedom
Degrees of freedom (df) is a concept that describes the number of values in a calculation that are free to vary. Understanding degrees of freedom is vital when working with t-distributions and confidence intervals.The formula to calculate degrees of freedom varies depending on the analysis. For a simple t-test, df is calculated as the sample size minus one, i.e., \( n - 1 \).Here's why degrees of freedom are important:
  • The shape of the t-distribution changes with different degrees of freedom.
  • A smaller sample (fewer degrees of freedom) leads to a wider distribution, affecting the critical value.
  • More degrees of freedom result in a t-distribution that resembles a normal distribution.
In the context of confidence intervals, knowing the degrees of freedom helps you find the correct t-value from the t-distribution table.
Alpha Level
The alpha level, often denoted as \( \alpha \), is a threshold that determines the level of significance in statistical tests and confidence interval calculations. It represents the probability of rejecting the null hypothesis when it is true.In terms of confidence intervals, the alpha level helps determine the range of the interval. A complementary concept to boundaries set by confidence levels:
  • The alpha level is calculated by subtracting the confidence level from 100%. For instance, a 95% confidence level corresponds to an alpha level of 0.05.
  • The smaller the alpha level, the more likely we are to include the true population parameter in the confidence interval.
  • In one-sided tests, the alpha level is directly used to find the critical region on the t-distribution.
Alpha levels are critical for determining how strict the confidence interval bounds are.
Critical Value
The critical value is an essential element in constructing confidence intervals and conducting hypothesis tests. It serves as a cutoff point, where if the test statistic exceeds this value, the null hypothesis is rejected. For t-distributions, the critical value depends on both the alpha level and degrees of freedom:
  • It is derived from the t-distribution table, taking into account specific df and 伪 values.
  • A higher confidence level will require a larger critical value, leading to a wider confidence interval.
  • The critical value marks the boundary of the confidence interval, ensuring that the interval has the desired level of confidence (95%, 99%, etc.).
The critical value helps define the range in which the true population parameter is expected to fall.

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Most popular questions from this chapter

An Izod impact test was performed on 20 specimens of PVC pipe. The sample mean is \(\bar{x}=1.25\) and the sample standard deviation is \(s=0.25 .\) Find a \(99 \%\) lower confidence bound on Izod impact strength.

An article in Engineering Horizons (Spring \(1990,\) p. 26 ) reported that 117 of 484 new engineering graduates were planning to continue studying for an advanced degree. Consider this as a random sample of the 1990 graduating class. (a) Find a \(90 \%\) confidence interval on the proportion of such graduates planning to continue their education. (b) Find a \(95 \%\) confidence interval on the proportion of such graduates planning to continue their education. (c) Compare your answers to parts (a) and (b) and explain why they are the same or different. (d) Could you use either of these confidence intervals to determine whether the proportion is actually \(0.25 ?\) Explain your answer.

Following are two confidence interval estimates of the mean \(\mathrm{m}\) of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the failure). $$ 3124.9 \leq \mu \leq 3215.7 \quad 3110.5 \leq \mu \leq 3230.1 $$ (a) What is the value of the sample mean cycles to failure? (b) The confidence level for one of these CIs is \(95 \%\) and for the other is \(99 \%\). Both CIs are calculated from the same sample data. Which is the \(95 \%\) CI? Explain why.

Suppose that \(X_{1}, X_{2}, \ldots, X_{\mathrm{n}}\) is a random sample from a continuous probability distribution with median \(\tilde{\mu}\) (a) Show that \(p\left\\{\min \left(X_{i}\right)<\tilde{\mu}<\max \left(X_{i}\right)\right\\}=1-\left(\frac{1}{2}\right)^{n-1}\) [Hint: The complement of the event \(\left[\min \left(X_{i}\right)<\tilde{\mu}\right.\) \(\left.<\max \left(X_{i}\right)\right]\) is \(\left[\max \left(X_{i}\right) \leq \tilde{\mu}\right] \cup\left[\min \left(X_{i}\right) \leq \tilde{\mu}\right]\) but \(\max\) \(\left(X_{i}\right) \leq \tilde{\mu}\) if and only if \(X_{i} \leq \tilde{\mu}\) for all \(\left.i .\right]\) (b) Write down a \(100(1-\alpha) \%\) confidence interval for the median \(\tilde{\mu}\) where $$ \alpha=\left(\frac{1}{2}\right)^{n-1} $$

Information on a packet of seeds claims that \(93 \%\) of them will germinate. Of the 200 seeds that I planted, only 180 germinated. (a) Find a \(95 \%\) confidence interval for the true proportion of seeds that germinate based on this sample. (b) Does this seem to provide evidence that the claim is wrong?
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