Let's analyze and address the questions based on the data and histogram provided.

Data Summary:

The total blood cholesterol levels (in mg/dL) for 20 adults are provided as:

• Measurements: 135, 138, 157, 159, 165, 166, 177, 177, 177, 187, 189, 192, 197, 197, 203, 212, 218, 221, 237, 241

From the histogram:

• The cholesterol levels are grouped into intervals with frequencies:
  • 125-150: 2 adults
  • 150-175: 4 adults
  • 175-200: 8 adults
  • 200-225: 4 adults
  • 225-250: 2 adults

Now, let's tackle the questions:

(a) Which measures of central tendency do not exist for this data set?

• Mean: Always exists for a data set.
• Median: Always exists.
• Mode: This represents the most frequent value. Since 177 appears three times (as the most frequent value), mode exists.

Answer:

• All of these measures exist.

(b) Suppose that the measurement 241 (the largest measurement in the data set) were replaced by 348. Which measures of central tendency would be affected by the change?

• Mean: Since the mean is sensitive to every data point, it will be affected by this change.
• Median: The middle value of the ordered data set remains unaffected by changes at the extremes, so median will not change.
• Mode: Mode depends on the most frequent values and doesn't change unless that specific value is changed, so mode will not be affected.

Answer:

• Mean

(c) Suppose that, starting with the original data set, the largest measurement were removed. Which measures of central tendency would be changed?

• Mean: Removing the largest value will change the total sum and number of observations, so the mean will be affected.
• Median: Since the middle values of the ordered list may not change drastically when the largest value is removed, median is less likely to change.
• Mode: The mode depends on the most frequent value, so it will remain unchanged.

Answer:

• Mean

(d) Which of the following best describes the distribution of the original data?

• A positively skewed distribution is when most of the data clusters at the lower end, and the tail extends towards the higher values. Given that the histogram has most of its frequencies at lower cholesterol levels (with a tail to the right), this data is positively skewed.

Answer:

• Positively skewed

Would you like further clarification or details on any of these answers? Here are 5 related questions you might consider:

1. How do you calculate the mean for a data set like this?
2. Can the median change if multiple values are altered at the extremes?
3. How does one calculate the mode in a more complex data set with multiple frequent values?
4. What are the characteristics of a negatively skewed distribution?
5. How does removing multiple data points affect the mean and median?

Tip: Always visualize data through histograms or box plots to better understand its distribution before calculating central tendencies.