Confidence Intervals, Point Estimates and  Sample Sizes

Finding and Interpreting Point Estimates of the Population Mean

The point estimate of a population mean is a single value that is used to estimate or approximate the average value of a population. It is calculated based on a sample taken from the population.

To find the point estimate of a population mean:

1. Take a random sample from the population of interest.

2. Calculate the mean of the sample, which is the average of all the values in the sample.

3. Use the sample mean as the point estimate of the population mean.

The idea behind using a sample to estimate the population mean is that if the sample is representative of the population, the sample mean should be a good approximation of the population mean. The larger the sample size, the more reliable the point estimate tends to be.

For example, if you want to estimate the average height of all adults in a country, you can randomly select a sample of adults, measure their heights, calculate the mean height of the sample, and use that mean as the point estimate for the population mean height.

It's important to note that a point estimate is just an estimate, and it may not be exactly equal to the true population mean. To quantify the precision or uncertainty of the estimate, it is common to also calculate a confidence interval, which provides a range of values within which the population mean is likely to fall.

Finding the Confidence Interval when the Population Standard Deviation is Known:  Guide, and Example.

Guide: 

1. Determine the desired confidence level: Specify the desired level of confidence, such as 95% or 90%. This represents the probability that the confidence interval will contain the true population mean.

2. Identify the sample mean: Take a random sample from the population and calculate the mean of the sample.

3. Find the population standard deviation (σ): If the population standard deviation is already known, you can use that value directly. If not, you will need to estimate it using the sample standard deviation (s) from the sample.

4. Determine the critical value: The critical value corresponds to the desired confidence level and the sample size. Look up the critical value from a Z-table or use a Z-score calculator. The critical value is often denoted as Zα/2, where α is the significance level (1 - confidence level) and /2 accounts for the two tails of the normal distribution.

5. Calculate the margin of error: The margin of error represents the range within which the population mean is likely to fall. It is calculated by multiplying the critical value (Zα/2) by the standard deviation (σ) divided by the square root of the sample size (n). The formula is: Margin of Error = Zα/2 * (σ / √n).

6. Calculate the confidence interval: Take the sample mean and subtract the margin of error to find the lower bound of the confidence interval. Then, take the sample mean and add the margin of error to find the upper bound of the confidence interval. The formula is: Confidence Interval = Sample Mean ± Margin of Error.

The resulting confidence interval represents the range of values within which we are confident (at the specified confidence level) that the true population mean lies.

Remember, when the population standard deviation is known, you can use the Z-distribution to find the critical value. If the population standard deviation is unknown, you would use the t-distribution instead, and the calculation would involve the sample standard deviation instead of the population standard deviation.

Example:

A sample of 260 observations is selected from a normal population with a population standard deviation of 26. The sample mean is 15. 

Determine the 98% confidence interval for the population mean. (Round your answers to 3 decimal places.) 

Finding the Confidence Interval when the Population Standard Deviation is Unknown Description: Guide and Example.

Guide: 

1. Determine the desired confidence level: Specify the desired level of confidence, such as 95% or 90%. This represents the probability that the confidence interval will contain the true population mean.

2. Take a random sample from the population: Collect a sample of data from the population of interest.

3. Calculate the sample mean (x̄) and sample standard deviation (s): Find the mean and standard deviation of the sample.

4. Determine the critical value: The critical value corresponds to the desired confidence level and the degrees of freedom (df). The degrees of freedom are equal to the sample size minus 1 (df = n - 1). Look up the critical value from a t-distribution table or use a t-score calculator.

5. Calculate the standard error: The standard error represents the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the sample standard deviation (s) by the square root of the sample size (n). The formula is: Standard Error = s / √n.

6. Calculate the margin of error: The margin of error represents the range within which the population mean is likely to fall. It is calculated by multiplying the critical value (tα/2) by the standard error. The formula is: Margin of Error = tα/2 * (s / √n).

7. Calculate the confidence interval: Take the sample mean (x̄) and subtract the margin of error to find the lower bound of the confidence interval. Then, take the sample mean and add the margin of error to find the upper bound of the confidence interval. The formula is: Confidence Interval = x̄ ± Margin of Error.

The resulting confidence interval represents the range of values within which we are confident (at the specified confidence level) that the true population mean lies, taking into account the uncertainty associated with the unknown population standard deviation.

Remember, when the population standard deviation is unknown, you use the t-distribution instead of the Z-distribution for finding critical values. This is because the t-distribution accounts for the additional uncertainty introduced by estimating the standard deviation from the sample.

Example: 

The German Beer Industry wants to estimate the mean yearly beer consumption. A sample of 19 people reveals the mean yearly consumption to be 70 gallons with a standard deviation of 14 gallons. Assume the population distribution is normal. 

Develop the 98% confidence interval for the population mean. (Round your answers to 3 decimal places.) 

Finding the Sample Proportion Guide and Example:

The sample proportion (p̂) represents the proportion or percentage of a certain characteristic or outcome within a sample. It is used to estimate the population proportion.

To find the sample proportion:

1. Take a random sample from the population of interest.

2. Determine the total number of observations in the sample (n).

3. Count the number of observations that have the desired characteristic or outcome (let's call this count x).

4. Calculate the sample proportion by dividing x by n: p̂ = x / n.

Here's a worked example:

Let's say you are interested in estimating the proportion of adults in a city who own a car. You take a random sample of 200 adults and find that 140 of them own a car.

1. Take a random sample of 200 adults.

2. The total number of observations in the sample is n = 200.

3. Count the number of adults who own a car, which is x = 140.

4. Calculate the sample proportion: p̂ = 140 / 200 = 0.7.

So, the sample proportion is 0.7, which means that 70% of the adults in the sample own a car. This sample proportion can be used to estimate the population proportion of car owners in the city.

Remember, the sample proportion is an estimate based on the sample data and is subject to sampling variability. It provides an approximation of the population proportion, but it may not be exactly equal to the true population proportion.

Finding the Confidence Interval for Population Proportion Description: Guide and Example

Guide:

1. Determine the desired confidence level: Specify the desired level of confidence, such as 95% or 90%. This represents the probability that the confidence interval will contain the true population proportion.

2. Take a random sample from the population: Collect a sample of data from the population of interest.

3. Determine the sample proportion (p̂): Count the number of observations in the sample that have the desired characteristic or outcome and divide it by the total sample size. This gives you the sample proportion.

4. Calculate the standard error: The standard error represents the standard deviation of the sampling distribution of the sample proportion. It is calculated by taking the square root of (p̂ * (1 - p̂) / n), where p̂ is the sample proportion and n is the sample size.

5. Determine the critical value: The critical value corresponds to the desired confidence level and is based on the normal distribution. For example, for a 95% confidence level, the critical value is approximately 1.96.

6. Calculate the margin of error: The margin of error represents the range within which the population proportion is likely to fall. It is calculated by multiplying the critical value by the standard error. The formula is: Margin of Error = Critical Value * Standard Error.

7. Calculate the confidence interval: Take the sample proportion and subtract the margin of error to find the lower bound of the confidence interval. Then, take the sample proportion and add the margin of error to find the upper bound of the confidence interval. The formula is: Confidence Interval = p̂ ± Margin of Error.

The resulting confidence interval represents the range of values within which we are confident (at the specified confidence level) that the true population proportion lies.

Remember, when finding the confidence interval for a population proportion, you assume that the sample is representative of the population, and the sampling distribution of the sample proportion is approximately normal.

Example:

Apple is considering a new sci-fi series. Before making a final decision, the studio managers design an experiment to estimate the proportion of viewers who would watch the series. A random sample of 920 viewers was selected and asked to watch the first two episodes. After viewing the episodes, 575 viewers indicated they would watch the new series.  

Develop a 99% confidence interval for the population proportion of people who would watch the new series. 

Determining the Necessary Sample Size to Estimate the Population Mean: Guide and Example.

Guide:

1. Determine the desired level of confidence: Specify the desired level of confidence, such as 95% or 90%. This represents the probability that the sample mean will fall within a certain margin of error of the true population mean.

2. Determine the desired margin of error: Specify the maximum allowable difference between the sample mean and the population mean. This is the margin of error.

3. Estimate the population standard deviation: If the population standard deviation is known, use that value. If not, estimate it based on previous data or pilot studies.

4. Use the formula: The formula to calculate the necessary sample size is: n = [(Z * σ) / E]^2, where n is the required sample size, Z is the Z-score corresponding to the desired level of confidence, σ is the estimated population standard deviation, and E is the desired margin of error.

5. Calculate the sample size: Plug in the values of Z, σ, and E into the formula to calculate the necessary sample size.

6. Round up: Since the sample size should be a whole number, round up to the nearest whole number to ensure an adequate sample size.

The resulting sample size represents the number of observations needed to estimate the population mean with the desired level of confidence and margin of error.

Remember, the larger the sample size, the smaller the margin of error and the higher the precision of the estimate. However, increasing the sample size also requires more resources and time, so it's important to strike a balance between accuracy and practical considerations.

Example:

A population's standard deviation is 8. We want to estimate the population mean with a margin of error of 1, with a 95% level of confidence. 

How large a sample is required? (Round your intermediate calculations to 2 decimal places and round up your answer to the next whole number.) 

Determining the Necessary Sample Size to Estimate the Population Proportion: Guide and Example.

Guide:

1. Determine the desired level of confidence: Specify the desired level of confidence, such as 95% or 90%. This represents the probability that the sample proportion will fall within a certain margin of error of the true population proportion.

2. Determine the desired margin of error: Specify the maximum allowable difference between the sample proportion and the population proportion. This is the margin of error.

3. Estimate the expected population proportion: Based on prior knowledge or pilot studies, estimate the expected population proportion.

4. Use the formula: The formula to calculate the necessary sample size is: n = [(Z^2 * p̂ * (1 - p̂)) / E^2], where n is the required sample size, Z is the Z-score corresponding to the desired level of confidence, p̂ is the estimated population proportion, and E is the desired margin of error.

5. Calculate the sample size: Plug in the values of Z, p̂, and E into the formula to calculate the necessary sample size.

6. Round up: Since the sample size should be a whole number, round up to the nearest whole number to ensure an adequate sample size.

The resulting sample size represents the number of observations needed to estimate the population proportion with the desired level of confidence and margin of error.

Remember, the larger the sample size, the smaller the margin of error and the higher the precision of the estimate. However, increasing the sample size also requires more resources and time, so it's important to strike a balance between accuracy and practical considerations.

Example:

The estimate of the population proportion should be within plus or minus 0.07, with a 98% level of confidence. The best estimate of the population proportion is 0.20. How large a sample is required?