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Estimation and Confidence Intervals - Point Estimation and Interval Estimation

Q: When you have no hypothesis to test, you have no idea what to expect in the population. Nevertheless, we can make estimations about the population, by using the sample distribution of means and its normality to construct a confidence interval around the found sample mean. What is a confidence interval?

A confidence interval is the interval or range of possible values within which an unknown population parameter is likely to be contained . E.g., You want to find out the average amount Dutch households spend each month on streaming service subscriptions. You have no idea how much it actually is, so you want to estimate the monthly amount by sampling 100 Dutch households. Using a 95% confidence interval, you want to identify the two values within which you are 95% confident that the average monthly spending of the Dutch households in the population on streaming service subscriptions lies.

Q: Confidence intervals can be calculated with a z-test when we know the population standard deviation ( σ ). This might seem counterintuitive, as knowing the population standard deviation ( σ ) suggests you would also know the population mean ( μ ),eliminating the need for estimation and allowing for factual statements instead. Can you explain in which cases you would still estimate the population mean ( μ ) despite knowing the population standard deviation ( σ )? In other words, in what scenarios might the population standard deviation ( σ ) be known while the population mean ( μ ) remains unknown?

Historical data: You know the σ , but the μ will have changed. The reason we an infer σ has not changed, is in situations where we have historical data for various dates in history and see that, every time the μ has changed, σ stayed fairly stable.

4 important questions on Estimation and Confidence Intervals - Point Estimation and Interval Estimation

When you have no hypothesis to test, you have no idea what to expect in the population.
Nevertheless, we can make estimations about the population, by using the sample distribution of means and its normality to construct a confidence interval around the found sample mean.
What is a confidence interval?

A confidence interval is the interval or range of possible values within which an unknown population parameter is likely to be contained.
E.g., You want to find out the average amount Dutch households spend each month on streaming service subscriptions.

You have no idea how much it actually is, so you want to estimate the monthly amount by sampling 100 Dutch households.
Using a 95% confidence interval, you want to identify the two values within which you are 95% confident that the average monthly spending of the Dutch households in the population on streaming service subscriptions lies.

Confidence intervals can be calculated with a z-test when we know the population standard deviation (σ).
This might seem counterintuitive, as knowing the population standard deviation (σ) suggests you would also know the population mean (μ),eliminating the need for estimation and allowing for factual statements instead.

Can you explain in which cases you would still estimate the population mean (μ) despite knowing the population standard deviation (σ)? In other words, in what scenarios might the population standard deviation (σ) be known while the population mean (μ) remains unknown?

Historical data: You know the σ, but the μ will have changed.

The reason we an infer σ has not changed, is in situations where we have historical data for various dates in history and see that, every time the μ has changed, σ stayed fairly stable.

What is the formula we can use to find the confidence intervals, when we are estimating the population mean, based on the distribution of sample means AND know the σ (population standard deviation)?

See formula.
Explanations of formula:

± means that µ is situated in between "M + (rest of formula)" and "M - (rest of formula)".

I.e., You have to calculate the formula with a "+" and with a "-".

The z score is dependent on how you have set ⍺.

E.g., if our confidence level is 95%, we look in the z-table for the z value where the area of c is 0.025 (2 sides so ⍺= 0.05 means 0.025 on both sides).
In that case our z would be 1.96.

What is the formula we can use to find the confidence intervals, when we are estimating the population mean, based on the distribution of sample means, in cases where we DON'T know the σ (population standard deviation)?

See formula.
Explanations of formula.

Sigma with accent: we don't know the population standard deviation, so we have to estimate it.
df: we need to know the degrees of freedom, because otherwise we can't find the right value in the t-table.
± means you have to find the upper value using '+' and the lower value using "-".

The question on the page originate from the summary of the following study material:

Statistics for the Behavioral Sciences | 9781483322254 | Gregory J Privitera

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