The breaking strengths (say, y) for one-foot- square samples of a particular synthetic fabric are approximately normally distributed with a mean of 2,280 pounds per square inch (psi) and a standard deviation of 10.6 psi.
a. Find the probability of selecting a 1-foot -square sample of material at random that on testing would have a breaking strength in excess of 2,240 psi.
b. Describe the sampling distribution for $\overline{Y}$ based on random samples of 25 one-foot sections. (you need to name the distribution and give the mean and standard deviation of the distribution)
Solution
Given that $\mu = 2280$ and $\sigma = 10.6$.
a. The probability of selecting a 1-foot -square sample of material at random that on testing would have a breaking strength in excess of 2,240 psi is
$$ \begin{aligned} P(Y > 2240) &= P(Y > 2240)\\ &= 1-P(Y < 2240)\\ &= 1-P\bigg(\frac{Y-\mu}{\sigma} < \frac{2240-2280}{10.6}\bigg)\\ &= 1-P(Z < -3.774)\\ &= 1-0.0001\\ &=0.9999 \end{aligned} $$
b. The sampling distribution for $\overline{Y}$ based on random samples of 25 one-foot sections is
$\overline{Y}\sim N(\mu,\sigma^2/n)$.
The sampling distribution of $\overline{Y}$ is normal distribution with mean $E(\overline{Y}) = \mu = 2280$ and standard deviation $\frac{\sigma}{\sqrt{n}} = \frac{10.6}{\sqrt{25}}= 2.12$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
