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The concept of standard error is crucial when dealing with statistics, especially in confidence intervals. It measures the variability or dispersion of a sample statistic—like the sample mean or proportion—from the population parameter. In simpler terms, it tells you how much the sample results might vary if you were to draw multiple samples from the same population.

The smaller the standard error, the less spread out the sample means or proportions are, indicating that the sample more accurately reflects the population.

• Standard error is used to account for the natural variance that can occur in sample data.
• It is calculated by dividing the standard deviation by the square root of the sample size.
• In this scenario, the standard error is given as 0.009, suggesting a relatively small variation in the estimate of the proportion of teenagers with hearing loss.

A small standard error in this context suggests that our sample proportion is a reliable estimate of the population proportion, meaning we can be reasonably confident in the interval we've created.

Normal distribution, also known as the bell curve, is a fundamental concept in statistics. It's a way to describe how the values of a variable are distributed. Most values tend to cluster around a central region, with values tapering off as they go further from the mean.

In a normal distribution:

• The mean, median, and mode of the distribution are all equal.
• It is symmetric about the mean.
• The total area under the curve is equal to 1.
• Distribution is described by two parameters: the mean (µ) and the standard deviation (σ).

For large enough sample sizes, the distribution of the sample means will tend to be normal, regardless of the population's initial distribution. This property is known as the Central Limit Theorem.
In the exercise, the sample size is large enough for the normal distribution to be valid, enabling us to make inferences about the population proportion using the bell curve. This is why using the normal distribution helps us construct the confidence interval.

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. Essentially, it tells you how far away a particular data point is from the mean of the data set.

• A Z-score of 0 indicates that the data point's score is identical to the mean.
• A positive Z-score indicates the data point is above the mean, while a negative score shows it is below the mean.
• Z-score is crucial when determining probabilities and identifying outliers in a data set.

In the context of confidence intervals, the Z-score corresponds to a specific confidence level. For a 90% confidence interval, we require a Z-score that captures the middle 90% of the data, leaving 5% in each tail of the normal distribution.
By looking up the standard normal distribution table, we find that a 90% confidence level corresponds to a Z-score of approximately 1.645. This value is then used to calculate the margins of error in our confidence interval formula, ensuring that our interval has the desired level of confidence.