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Chapter 7: Problem 34

A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of \(8.48 \mathrm{MPa}\) and a sample standard deviation of .79 MPa (" Characterization of Bearing Strength Factors in Pegged Timber Connections," J. of Structural Engr, 1997:326-332). a. Calculate and interpret a \(95 \%\) lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress? b. Calculate and interpret a \(95 \%\) lower prediction bound for the proportional limit stress of a single joint of this type.

Short Answer

Expert verified
95% confidence that the true average stress > 8.11 MPa and a single joint stress > 6.99 MPa, assuming normality.

Step by step solution

01

Identify the Problem

We need to find a 95% lower confidence bound for the mean and a 95% lower prediction bound for a single observation from a sample. The sample mean is 8.48 MPa, the standard deviation is 0.79 MPa, and the sample size is 14.
02

Step 2a: Calculate the Standard Error of the Mean

The standard error (SE) is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} \] where \(s = 0.79\) MPa and \(n = 14\). Thus, \[ SE = \frac{0.79}{\sqrt{14}} = 0.211 \] MPa.
03

Step 2b: Determine the t-value for 95% Confidence

The 95% lower confidence bound requires the t-value from the t-distribution for \( n-1 = 13 \) degrees of freedom. For 95% confidence level in one tail (lower bound), it is approximately \(-1.771\).
04

Calculate the 95% Lower Confidence Bound

We use the formula for the lower confidence bound: \[ \bar{x} + t \cdot SE \] where \(\bar{x} = 8.48\) MPa, \(t = -1.771\), and \(SE = 0.211\). Calculate: \[ 8.48 - (1.771 \times 0.211) = 8.11 \] MPa.
05

Interpret the Confidence Bound

The 95% lower confidence bound means we are 95% confident that the true average proportional limit stress is greater than 8.11 MPa, assuming the data are normally distributed.
06

Step 5a: Calculate the Prediction Interval Standard Error

Prediction interval SE incorporates both the sample standard deviation and SE of mean: \[ SE_{pred} = s \sqrt{1 + \frac{1}{n}} \] Thus, \[ SE_{pred} = 0.79 \sqrt{1 + \frac{1}{14}} = 0.837 \] MPa.
07

Step 5b: Determine the t-value for 95% Prediction

Use the same t-value for confidence bound, \(-1.771\), assuming a normal distribution.
08

Calculate the 95% Lower Prediction Bound

Apply the bound formula: \[ \bar{x} + t \cdot SE_{pred} \] where \( \bar{x} = 8.48 \), \( t = -1.771 \), \( SE_{pred} = 0.837 \) MPa. Calculate: \[ 8.48 - (1.771 \times 0.837) = 6.99 \] MPa.
09

Interpret the Prediction Bound

The 95% lower prediction bound means we are 95% confident that for a single joint, the proportional limit stress is greater than 6.99 MPa, assuming normal distribution.

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Key Concepts

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

Confidence Intervals
Confidence intervals are essential in statistical inference. They give us a range within which we expect the true population parameter to lie, based on our sample data. For example, if we calculate a 95% confidence interval for the mean stress of joints, it suggests that we are 95% confident that the true mean falls within that range.

There are some key points about confidence intervals:
  • The confidence level (like 95%) tells us how sure we are about this interval capturing the true mean.
  • We use the sample mean and standard deviation to estimate these intervals.
  • The width of the interval depends on both the variability in the data and the sample size.
  • A larger sample size can lead to a narrower interval, providing a more precise estimate.
In this exercise, we calculated a lower confidence bound, a variation focusing on one side of the range. This tells us that the true mean is likely above a certain point, in this case, 8.11 MPa.
t-Distribution
The t-distribution is crucial for calculating confidence intervals when the sample size is small, typically less than 30 samples. It adjusts for the increased variability expected in smaller samples. Unlike the normal distribution, which assumes we know the true population standard deviation, the t-distribution uses the sample standard deviation, making it more adaptable.

Key features of the t-distribution include:
  • It is similar to the normal distribution but has heavier tails, allowing for more variability.
  • As the sample size increases, the t-distribution approaches the normal distribution.
  • The degrees of freedom (n-1 for a sample size of n) help determine its shape.
In this exercise, with 13 degrees of freedom, we used a t-value of approximately -1.771 to calculate the lower confidence and prediction bounds.
Prediction Bounds
Prediction bounds take confidence intervals a step further by estimating the range for individual future observations. Instead of focusing on a parameter like the mean, a prediction bound addresses the variability of single data points, accounting for both sample-based variability and individual observation variability.

Key characteristics of prediction bounds include:
  • They are wider than confidence intervals because they need to cover more uncertainty.
  • They take into account the sample mean's standard error and additional variability for individual predictions.
  • The sample size and variability are crucial for determining their width.
For example, the 95% lower prediction bound of 6.99 MPa tells us that an individual joint's stress is highly likely to exceed this value.
Sample Mean
The sample mean is a fundamental concept in statistics. It represents the average value of all data points in a sample. In this exercise, the sample mean of 8.48 MPa serves as a point estimate for the true population mean.

Here鈥檚 what to know about sample means:
  • They are calculated by summing all sample values and dividing by the number of observations.
  • The sample mean is our best estimate of the population mean.
  • If the sample is well-chosen, the sample mean provides a reliable estimate of the population mean.
While the sample mean is useful, it鈥檚 only a starting point. With a small sample, like 14 specimens, further analysis with standard errors and prediction bounds is necessary to understand its accuracy.
Standard Error
The standard error (SE) measures the variability of the sample mean estimate, indicating how much the sample mean might fluctuate from the true population mean. It鈥檚 a crucial part of confidence intervals and prediction bounds.

Important aspects of standard errors include:
  • It's calculated as the sample standard deviation divided by the square root of the sample size, \( SE = \frac{s}{\sqrt{n}} \).
  • A smaller SE suggests a more precise estimate of the population mean.
  • As the sample size grows, the SE decreases, implying better estimation.
In this exercise, the SE was 0.211 MPa, helping determine the width of the confidence and prediction bounds, enhancing our prediction and confidence in the results.

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Most popular questions from this chapter

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