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The image you've provided contains several statistics questions based on a dataset and a histogram. Let's go through the questions one by one and analyze the data.

### The dataset:
The dataset is a set of time durations (in minutes) for 22 people completing an intelligence test. The respective times are:

23, 29, 31, 31, 32, 34, 36, 37, 38, 39, 39, 39, 40, 39, 42, 42, 42, 42, 43, 44, 44, 44

The histogram provided shows the frequency distribution of the data.

Now let's go through the questions and answer them:

### (a) Which measures of central tendency do exist for this data set? Choose all that apply.
The possible answers are:
- Mean
- Median
- Mode
- All of the measures exist

**Answer**: 
- **All of the measures exist** (since we can calculate the mean, median, and mode from the data).

### (b) Suppose that the measurement 23 (the smallest measurement in the data set) were replaced by 9. Which measures of central tendency would be affected by the change? Choose all that apply.

When 23 is replaced by 9, the following will happen:
- **Mean**: Yes, the mean will be affected since it is sensitive to every value in the data.
- **Median**: The median may not be affected if the replacement of the smallest value does not change the middle of the data.
- **Mode**: No, mode won't be affected since the most frequent values remain unchanged.

**Answer**:
- **Mean**

### (c) Suppose that, starting with the original data set, the smallest measurement were removed. Which measures of central tendency would be changed from those of the original data set? Choose all that apply.

If the smallest measurement (23) is removed, the following happens:
- **Mean**: Yes, it would change because removing a value changes the average.
- **Median**: It might change because removing an extreme value can shift the central value.
- **Mode**: No, mode will not change since the most frequent values stay the same.

**Answer**:
- **Mean**
- **Median**

### (d) Which of the following best describes the distribution of the original data? Choose only one.
The possible answers are:
- Negatively skewed
- Positively skewed
- Roughly symmetrical

Based on the histogram and the dataset, the distribution has a longer tail on the lower end (toward the smaller values like 23), which indicates **positive skewness** (i.e., skewed to the right).

**Answer**: 
- **Positively skewed**

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Let me know if you would like detailed calculations for any part of this! Here are some related questions to deepen your understanding:

1. How would you calculate the exact mean of this dataset?
2. What steps are involved in finding the median in this data?
3. Could removing multiple values from the dataset affect all measures of central tendency?
4. How would the distribution look if the highest values were removed instead?
5. What is the effect of outliers on the mean compared to the median?

### Tip:
Remember, the mean is sensitive to every single value in the dataset, while the median is more resistant to outliers and extreme values.

An intelligence test with a maximum completion time of 45 minutes was administered to a group of 22 people. Their respective times for completion (in minutes) were: 23, 29, 31, 31, 32, 34, 36, 37, 38, 39, 39, 39, 40, 39, 42, 42, 42, 42, 43, 44, 44, 44. Several questions are posed about the central tendency and distribution.

Analyze the central tendency (mean, median, mode) and skewness in a dataset of test completion times. Explore how changing or removing values affects these statistical measures. Suitable for Grades 9-12.

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