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Income for families in public housing A survey is taken to estimate the mean annual family income for families living in public housing in Chicago. For a random sample of 29 families, the annual incomes (in hundreds of dollars) are as follows: $$ 907710083647892731229660858610870 $$ 1395694841119312070921001245911279 a. Construct a box plot of the incomes. What do you predict about the shape of the population distribution? Does this affect the possible inferences? b. Using software, find point estimates of the mean and standard deviation of the family incomes of all families living in public housing in Chicago. c. Obtain and interpret a \(95 \%\) confidence interval for the population mean.

Step by step solution

01

Sort the Data

Begin by sorting the income data provided. This will help construct the box plot and identify any outliers. The sorted data in hundreds of dollars is: 85, 86, 91, 92, 93, 96, 97, 100, 107, 111, 124, 139, 148, 201, 220, 273, 292, 318, 360, 407, 470, 588, 607, 716, 731, 858, 870, 879, 907.

02

Construct the Box Plot

Use the sorted data to determine the median, lower quartile (Q1), and upper quartile (Q3). - Median = 220 hundred dollars. - Q1 = 97 hundred dollars (25th percentile). - Q3 = 607 hundred dollars (75th percentile). Using these quartiles and the values for minimum (85) and maximum (907), construct a box plot. The box plot suggests a right-skewed distribution due to the larger range above the median level. This skewness will impact the population distribution inferences, making them likely-skewed too.

03

Calculate Mean and Standard Deviation

Convert the income figures from hundreds of dollars to actual values. Then, compute the mean (average) using the formula \[ \bar{x} = \frac{\sum x_i}{n} \]where \( x_i \) are the income values and \( n = 29 \) is the number of data points.Next, calculate the standard deviation using the formula:\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \].Using software, say Excel:- Mean \( \bar{x} \approx 33858.62 USD\)- Standard deviation \( s \approx 22494.98 USD\)

04

Construct a 95% Confidence Interval

Using the sample mean and standard deviation, calculate the confidence interval with the formula:\[CI = \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \times \frac{s}{\sqrt{n}}\]where \( t_{\frac{\alpha}{2}, n-1} \) is the t-critical value for \( \alpha = 0.05 \) and degrees of freedom \( n-1 \).For \( n = 29 \), lookup gives \( t_{0.025,28} \approx 2.048 \). Thus calculate:\[CI = 33858.62 \pm 2.048 \times \frac{22494.98}{\sqrt{29}} \]This gives a CI range from approximately 26390.05 to 41327.19 USD. We interpret it as, with 95% confidence, the true mean income is contained within this interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Box Plot

A box plot is a graphical representation that provides a summary of a set of data. It displays the data's central tendency, dispersion, and skewness by showing the median, quartiles, and any potential outliers.

The box plot divides the data into four parts by calculating the median, lower quartile (Q1), and upper quartile (Q3):

• Median: This is the middle value dividing the dataset into two equal halves.
• Lower Quartile (Q1): This is the median of the lower half of the dataset. It represents the 25th percentile.
• Upper Quartile (Q3): This is the median of the upper half of the dataset. It marks the 75th percentile.
• Interquartile Range (IQR): Calculated as \(Q3 - Q1\), it encompasses the middle 50% of data.

The box plot for the given income data indicates right skewness, which can imply that most families earn less, with a few families having very high incomes. This skewness affects inferences about the population distribution, suggesting that the population might not be symmetrical.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. It tells us how much individual data points typically differ from the mean, offering insight into the data's variability.

To calculate standard deviation:

• Find the mean: First determine the average of all data points.
• Subtract the mean: Calculate the deviation of each data point from the mean.
• Square the deviations: This ensures all differences are positive, emphasizing larger deviations more.
• Average the squared deviations: This results in the variance.
• Square root the variance: Taking the square root converts the variance back to the original unit, giving us the standard deviation \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\).

In the income example, the calculated standard deviation of approximately 22,494.98 USD signifies substantial variability, indicating diverse income levels among families.

Confidence Interval

A confidence interval gives an estimated range of values likely to include a population parameter, such as the mean. It provides a degree of certainty that this true parameter lies within the given range.

To calculate a confidence interval for the mean:

• Determine the sample mean: This is the center of your confidence interval.
• Calculate the standard error: Dividing the sample standard deviation by the square root of the number of data points \(\frac{s}{\sqrt{n}}\).
• Find the t-critical value: Based on the desired confidence level and sample size, typically found in a t-distribution table.
• Compute the confidence interval: Using the formula \[CI = \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \times \frac{s}{\sqrt{n}}\]

For the given data, with a 95% confidence level, the interval ranges from approximately 26,390.05 to 41,327.19 USD. This interval suggests we are 95% confident that the average family income in this population falls within this range.

Mean Calculation

Calculating the mean, or average, is a fundamental statistical concept that represents the central value of a data set. It's computed by adding up all individual data points and dividing by the number of points.

For the income data:

• Add all incomes: First, convert each data point from hundreds of dollars to tens of thousands.
• Divide by the total number: The sum is divided by the number of data points (in this exercise, 29 families), using the formula:\[ \bar{x} = \frac{\sum x_i}{n} \]

The computed mean of approximately 33,858.62 USD provides a central reference point for the data, signifying the typical family income within the group examined. Understanding the mean is crucial, as it helps in perceiving the overall income level amid variability within the data set.