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Answer :
Sure! Let's break down the solution to find the various statistics and conduct the Q test for the given dataset: 116.0, 97.9, 114.2, 106.8, and 108.3.
1. Mean:
- To find the mean, sum up all the numbers and divide by the count of the numbers.
- [tex]\(\text{Mean} = \frac{116.0 + 97.9 + 114.2 + 106.8 + 108.3}{5} = 108.64\)[/tex]
2. Standard Deviation:
- Standard deviation measures the dispersion of data from the mean. We're using the sample standard deviation formula here.
- The standard deviation of the dataset is approximately 7.14.
3. Median:
- To find the median, arrange the numbers in ascending order and find the middle number.
- The sorted data is: 97.9, 106.8, 108.3, 114.2, 116.0.
- Since there are 5 numbers, the third number is the median: 108.3.
4. Range:
- The range is the difference between the maximum and minimum values in the dataset.
- [tex]\(\text{Range} = 116.0 - 97.9 = 18.1\)[/tex]
5. 90% Confidence Interval for the Mean:
- A confidence interval gives an estimated range of values which is likely to include the mean with a certain level of confidence.
- For this dataset, the 90% confidence interval for the mean is approximately (101.83, 115.45).
6. Q Test for Outliers:
- The Q test can help determine if a particular data point is an outlier.
- Identify the gap (difference) between the suspected outlier (97.9) and the closest value (106.8). The gap is 106.8 - 97.9 = 8.9.
- The range of the dataset is still 18.1.
- Calculate the Q value as the gap divided by the range: [tex]\( \frac{8.9}{18.1} \approx 0.49 \)[/tex]
- Compare this Q value with the critical Q value for a dataset of 5 points at a 90% confidence level, which is 0.679.
- Since 0.49 is less than 0.679, the value 97.9 should not be discarded as an outlier.
This step-by-step breakdown provides a comprehensive understanding of how each statistical measure and test is calculated for the dataset in question.
1. Mean:
- To find the mean, sum up all the numbers and divide by the count of the numbers.
- [tex]\(\text{Mean} = \frac{116.0 + 97.9 + 114.2 + 106.8 + 108.3}{5} = 108.64\)[/tex]
2. Standard Deviation:
- Standard deviation measures the dispersion of data from the mean. We're using the sample standard deviation formula here.
- The standard deviation of the dataset is approximately 7.14.
3. Median:
- To find the median, arrange the numbers in ascending order and find the middle number.
- The sorted data is: 97.9, 106.8, 108.3, 114.2, 116.0.
- Since there are 5 numbers, the third number is the median: 108.3.
4. Range:
- The range is the difference between the maximum and minimum values in the dataset.
- [tex]\(\text{Range} = 116.0 - 97.9 = 18.1\)[/tex]
5. 90% Confidence Interval for the Mean:
- A confidence interval gives an estimated range of values which is likely to include the mean with a certain level of confidence.
- For this dataset, the 90% confidence interval for the mean is approximately (101.83, 115.45).
6. Q Test for Outliers:
- The Q test can help determine if a particular data point is an outlier.
- Identify the gap (difference) between the suspected outlier (97.9) and the closest value (106.8). The gap is 106.8 - 97.9 = 8.9.
- The range of the dataset is still 18.1.
- Calculate the Q value as the gap divided by the range: [tex]\( \frac{8.9}{18.1} \approx 0.49 \)[/tex]
- Compare this Q value with the critical Q value for a dataset of 5 points at a 90% confidence level, which is 0.679.
- Since 0.49 is less than 0.679, the value 97.9 should not be discarded as an outlier.
This step-by-step breakdown provides a comprehensive understanding of how each statistical measure and test is calculated for the dataset in question.
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