91Ó°ÊÓ

When working with data, the sample proportion is a critical concept, especially in statistics involving categories like success or failure. In this case, the success was defined as a mask's lens popping out at a specific temperature. The sample proportion, denoted as \( \hat{p} \), helps in estimating how common this event is within the population.

To calculate the sample proportion, count the number of successes (lenses popping out) and divide by the total number of observations (masks tested). For our test:

• Number of successes = 11
• Total number of observations = 55
• Sample Proportion, \( \hat{p} = \frac{11}{55} = 0.2 \)

This means that 20% of the tested masks had their lenses pop out at 250°F.

A sample proportion like this one is essential for constructing a confidence interval, which provides insight into what the true proportion in the entire population might be.

The Z-score is a statistical metric that indicates how many standard deviations an element is from the mean of its distribution. In the context of confidence intervals, it helps determine the range of values within which we expect the true population proportion to lie.

For a 90% confidence interval, we use a Z-score of 1.645. But why 1.645? A 90% confidence level implies there is a 5% chance in each of the two tails of the standard normal distribution that the true proportion can fall outside our interval.

To find this, we locate the value on the Z-distribution table corresponding to the central 90% of the distribution, which leaves 5% in the tails on each side. This means:

• Lower tail: 5%
• Upper tail: 5%
• Central area covered: 90%

The Z-score marks the point dividing the bottom 95% from the top 5%, which is 1.645.

Standard error is an important concept when linking the sample data and potential population data. It measures the amount of variability or "spread" we might expect in the sample proportions if we took numerous samples from the same population.

To calculate the standard error of the sample proportion \( \hat{p} \), we use the formula:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\] where:

• \( \hat{p} \) is the sample proportion,
• \( n \) is the sample size.

So, with our numbers:

• \( \hat{p} = 0.2 \)
• \( n = 55 \)
• \( SE = \sqrt{\frac{0.2 \times 0.8}{55}} \approx 0.0537 \)

This standard error of approximately 0.0537 suggests how much the sample proportion might differ from the true population proportion. This variability is used, along with the Z-score, to construct a confidence interval, providing us with a range that likely contains the true proportion.