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The standard error (SE) is a key statistical concept used to measure the precision of the sample mean as an estimate of the population mean.
It tells us how much variability we can expect between different samples drawn from the same population.
The standard error is calculated using the formula:

• SE = \(\frac{s}{\sqrt{n}}\)

Where:

• \(s\) is the sample standard deviation.
• \(n\) is the sample size.

By dividing the standard deviation by the square root of the sample size, we adjust for larger sample sizes, acknowledging that more data will tend to give a more precise estimate of the population mean.
A smaller standard error indicates that the sample mean is a more accurate reflection of the population mean.

A Z-Score is a statistical measure that describes a value's position relative to the mean of a group of values.
In simpler terms, it tells us how many standard deviations away a data point is from the mean.
For constructing confidence intervals, the Z-score corresponds to the amount of data within a certain level of confidence. In our scenario, a 90% confidence interval leaves 5% in each tail of a standard normal distribution.
This is because 100% minus 90% leaves 10%, with 5% in each tail.
To find the Z-score, you can consult a Z-table or use a calculator, where for 90% confidence, the Z-score is typically 1.645.
This Z-score helps in determining the range in which we expect to find the true population mean with 90% confidence.

The margin of error (E) gives us an idea of how accurate our confidence interval is, or how much we are likely to be wrong.
It is computed as follows:

• E = Z * SE

Where:

• Z is the Z-score found for the desired confidence level.
• SE is the standard error of the sample mean.

The margin of error reflects uncertainty, showing how far off the estimate of the population mean might be.
A smaller margin of error indicates a more reliable confidence interval.
It essentially adds and subtracts around the sample mean to establish the interval lower and upper bounds.

The population mean (\( \mu \)) is the average of a set of characteristics for the entire population we are interested in.
However, since calculating the population mean directly is often challenging due to the size of populations, we use sample data to estimate it.In a survey like ours, the sample mean provides an estimate of the population mean.
By using confidence intervals, we get a range that is likely to contain the true population mean.
With a 90% confidence interval, there is a 90% chance the interval includes the true population mean.
This means we accept a 10% risk of the interval not capturing \( \mu \). This statistical estimation ensures that conclusions drawn about population means are backed by a degree of certainty.