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A sample proportion is a simple way to estimate the proportion of a certain characteristic in a population. Think of it like getting a tiny snapshot of a bigger picture. In our case with ceramic hips, the sample proportion tells us how many of those hips are actually squeaking.
In this exercise, we find the sample proportion by dividing the number of squeaking hips by the total hips in the study.

• Number of squeaking hips: 10
• Total hips: 143

So, the sample proportion is calculated as:\[ \hat{p} = \frac{10}{143} \approx 0.0699 \]This means that approximately 6.99% of the ceramic hips in the sample study developed a squeak. By calculating this proportion, we create a starting point for further statistical analysis, allowing us to draw conclusions about the larger population of hip implants based on our sample.

The standard error (SE) gives us an idea about the variability or spread of our sample proportion if we took many samples. It's basically telling us how much we can expect our sample proportion to fluctuate from the true proportion in the entire population.
For calculating the standard error, we use the formula:\[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]In our exercise:

• \( \hat{p} = 0.0699 \)
• \( n = 143 \)

This gives:\[ SE \approx \sqrt{ \frac{0.0699 \times 0.9301}{143} } \approx 0.0217 \]The standard error of about 0.0217 indicates how much the sample proportion could vary from the true population proportion by just considering the chance variation in sampling.

A Z-score is a measure that tells us how many standard deviations an element is from the mean. In the context of confidence intervals, it helps determine how certain we are about our estimates.
For a one-sided confidence interval at a 95% confidence level, we utilize a Z-score of 1.645. This value comes from standard normal distribution tables, which tell us the probability of certain values occurring.

• A Z-score of 1.645 means that we expect our sample estimate to fall within a range that covers 95% of all possible outcomes just by pure chance.

By using a Z-score, we can build a boundary around our sample proportion to estimate where the true population proportion might be.

The lower confidence bound provides a boundary or limit where we can say with a certain level of confidence that the true proportion lies above it. It's like saying, "I am sure that the proportion is at least this high."
To find this lower confidence bound, we use the formula:\[ \hat{p} - Z \times SE \]Substituting our values:

• \( \hat{p} = 0.0699 \)
• \( Z = 1.645 \)
• \( SE = 0.0217 \)

Yields:\[ 0.0699 - 1.645 \times 0.0217 \approx 0.0343 \]Therefore, at a 95% confidence level, we can claim that the true proportion of squeaking hips in the larger population is at least 3.43%. This bound provides us with a level of assurance that, given our data, the true squeaking rate isn’t likely lower than this calculated figure.