91Ó°ÊÓ

In statistics, a difference of proportions refers to the difference between two independent sample proportions.
In this exercise, we focus on alloy A and alloy B, which have different proportions of specimens with at least 34 ksi in strength.
For alloy A, the proportion is calculated as \( \hat{p}_A = \frac{22}{40} = 0.55 \), and for alloy B, it is \( \hat{p}_B = \frac{29}{42} \approx 0.6905 \).
The difference of proportions \((\hat{p}_A - \hat{p}_B)\) is calculated to understand how the strength distribution between the two alloys varies.
This difference is simply \( 0.55 - 0.6905 = -0.1405 \).
The negative difference suggests that a smaller proportion of alloy A specimens have at least 34 ksi compared to alloy B.
This calculation is fundamental in assessing whether this observed difference is likely due to random variation or a true difference.

The standard error (SE) of the difference between two proportions quantifies the variability of the difference observed in sample proportions.
It acts as a gauge of how precise our difference of proportions estimate is and helps construct the confidence interval.
The formula used is: \[ SE(\hat{p}_A - \hat{p}_B) = \sqrt{ \frac{\hat{p}_A(1 - \hat{p}_A)}{m_A} + \frac{\hat{p}_B(1 - \hat{p}_B)}{m_B} } \]This formula considers the sample sizes \(m_A = 40\) for alloy A and \(m_B = 42\) for alloy B.
By substituting the sample proportions, we have: \[ SE(\hat{p}_A - \hat{p}_B) \approx \sqrt{ \frac{0.55 \times 0.45}{40} + \frac{0.6905 \times 0.3095}{42} } \approx 0.1091 \]
A smaller SE means a more precise difference estimate, making the confidence interval narrower.

The critical value is a factor that determines the width of the confidence interval.
For a 95% confidence level, the critical value is typically 1.96 from the standard normal distribution (Z-distribution).
It indicates how many standard errors we need to go up and down from our estimated difference to capture the true difference with 95% confidence.

• A higher critical value would create a wider confidence interval, implying more uncertainty.
• Conversely, a lower critical value would lead to a narrower interval, suggesting more certainty about the estimate.

Using the critical value of 1.96 is standard practice for constructing confidence intervals when assuming normal distribution.

Frequency distribution is a way of summarizing data that shows how often each different value occurs.
In this exercise, ultimate strength values for specimens made from alloys A and B are categorized into intervals.

• From 26 to less than 30 ksi: 6 specimens for A, 4 for B.
• 30 to less than 34 ksi: 12 specimens for A, 9 for B.
• 34 to less than 38 ksi: 15 specimens for A, 19 for B.
• 38 to less than 42 ksi: 7 specimens for A, 10 for B.

This distribution helps in calculating the sample proportions, as it breaks the data into manageable categories, making it easier to identify the number of specimens meeting a specific criterion, such as having at least 34 ksi in strength.
Understanding and interpreting these distributions are vital in statistical analysis, as they convey crucial insights about the data’s overall pattern.