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Chapter 7: Problem 4

A CI is desired for the true average stray-load loss \(\mu\) (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of \(1500 \mathrm{rpm}\). Assume that strayload loss is normally distributed with \(\sigma=3.0\). a. Compute a \(95 \%\) CI for \(\mu\) when \(n=25\) and \(\bar{x}=58.3\). b. Compute a \(95 \%\) CI for \(\mu\) when \(n=100\) and \(\bar{x}=58.3\). c. Compute a \(99 \%\) CI for \(\mu\) when \(n=100\) and \(\bar{x}=58.3\). d. Compute an \(82 \%\) CI for \(\mu\) when \(n=100\) and \(\bar{x}=58.3\). e. How large must \(n\) be if the width of the \(99 \%\) interval for \(\mu\) is to be \(1.0\) ?

Short Answer

Expert verified
95% CI, n=25: [57.124, 59.476]; 95% CI, n=100: [57.712, 58.888]; 99% CI, n=100: [57.527, 59.073]; 82% CI, n=100: [57.898, 58.702]; n needed for 99% CI width 1.0: 239.

Step by step solution

01

Understand the Problem

We need to calculate the confidence intervals (CI) for the average stray-load loss \( \mu \) in an induction motor. The stray load loss is normally distributed with a known standard deviation \( \sigma = 3.0 \). We need to calculate confidence intervals for different sample sizes and confidence levels.
02

Compute 95% CI for n=25

For a 95% confidence interval, the Z-score is approximately 1.96. The sample mean \( \bar{x} \) is 58.3, \( n = 25 \). The formula for the confidence interval is:\[\bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}\]Plugging in the values, we get:\[58.3 \pm 1.96 \cdot \frac{3.0}{\sqrt{25}} = 58.3 \pm 1.176\]Thus, the 95% CI is \([57.124, 59.476]\).
03

Compute 95% CI for n=100

For \( n=100 \), using the same 95% confidence level (Z=1.96):\[\frac{3.0}{\sqrt{100}} = 0.3\]So, the confidence interval is:\[58.3 \pm 1.96 \cdot 0.3 = 58.3 \pm 0.588\]Thus, the 95% CI is \([57.712, 58.888]\).
04

Compute 99% CI for n=100

For a 99% confidence interval, the Z-score is approximately 2.576. With \( n=100 \):\[\frac{3.0}{\sqrt{100}} = 0.3\]The confidence interval is:\[58.3 \pm 2.576 \cdot 0.3 = 58.3 \pm 0.7728\]Thus, the 99% CI is \([57.5272, 59.0728]\).
05

Compute 82% CI for n=100

For an 82% confidence interval, the Z-score is approximately 1.34. For \( n=100 \):\[\frac{3.0}{\sqrt{100}} = 0.3\]The confidence interval is:\[58.3 \pm 1.34 \cdot 0.3 = 58.3 \pm 0.402\]Thus, the 82% CI is \([57.898, 58.702]\).
06

Determine Required n for 99% CI Width of 1.0

Use the formula for the width of the confidence interval:\[W = 2 \times Z \cdot \frac{\sigma}{\sqrt{n}}\]For a 99% CI, \( W = 1.0 \), and \( Z = 2.576 \). Rearrange to solve for \( n \):\[1.0 = 2 \times 2.576 \cdot \frac{3.0}{\sqrt{n}}\]\[\frac{1.0}{5.152} = \frac{3.0}{\sqrt{n}}\]\[\sqrt{n} = \frac{3.0 \times 5.152}{1.0}\]\[\sqrt{n} = 15.456\]\[n = 239\]Thus, a sample size of \( n = 239 \) is required for the desired width.

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Key Concepts

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

Normal Distribution
When we talk about normal distribution, we're referring to a way data tends to cluster around a central value. Think of it as a symmetrical, bell-shaped curve that represents how data is spread. This curve is pivotal in statistics, often used because it simplifies many kinds of analyses. When a set of data, like stray-load loss from induction motors, follows a normal distribution, it implies most of the values are close to the mean, and very few are extreme low or high values.
In a normal distribution:
  • The center of the curve represents the mean, median, and mode, which are all the same.
  • The spread of the curve is determined by the standard deviation, indicating the range of data.
  • Data points that lie far from the center are rare.
This concept is essential when constructing confidence intervals, as it forms the basis for assuming how data behaves.
Sample Size
Sample size, denoted as 'n' in statistics, is simply the number of observations in a sample. The sample size plays a crucial role in the accuracy and reliability of the statistical analysis.
Here's why it matters:
  • Larger sample sizes tend to produce more reliable results. This is because they better represent the overall population.
  • Small samples can lead to skewed results and may not reflect the true average or variations present.
  • The formula for calculating confidence intervals involves the square root of the sample size, which means increasing the sample size reduces the margin of error.
In the case of computing the confidence interval, a larger 'n' results in a narrower interval, implying more precision in estimating the population mean.
Z-score
Z-score is a critical concept when working with confidence intervals and normal distribution. It represents the number of standard deviations a data point is from the mean. In constructing confidence intervals, Z-scores help determine the boundaries of the interval.
Here鈥檚 how Z-scores fit in:
  • Each confidence level corresponds to a specific Z-score. For example, a 95% confidence level corresponds to a Z-score of approximately 1.96.
  • A Z-score allows us to standardize different distributions for comparison.
  • It helps depict how far data points are from the average value.
When calculating confidence intervals, like in our exercise, we use Z-scores to multiply with the standard error to reflect a range in which the true population parameter lies.
Standard Deviation
Standard deviation is a measure of how spread out the numbers are in a data set. In a normal distribution, it鈥檚 pivotal as it determines the shape of the bell curve.
  • A small standard deviation indicates that values tend to be very close to the mean.
  • A larger standard deviation indicates values are spread out over a wider range.
  • In the exercise, we have a known standard deviation (蟽) of 3.0, which aids in calculating the confidence interval.
When constructing confidence intervals, standard deviation is part of the calculation for the standard error (SE), which is crucial for determining the range of possible values for the population mean. The formula for standard error involves dividing the standard deviation by the square root of the sample size, highlighting how sample size and standard deviation together influence the confidence interval's bounds.

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