To determine the inferences based on the data provided about the number of pages Georgina read in books last summer and this summer, we need to analyze the measures of central tendency and variability. Here are the steps and the detailed, step-by-step analysis:
1. Calculate the mean (average) number of pages for each summer:
- Last Summer: 140, 220, 190, 180, 190
- [tex]\[\text{Mean of Last Summer} = \frac{140 + 220 + 190 + 180 + 190}{5} = 184.0\][/tex]
- This Summer: 150, 170, 150, 190, 180
- [tex]\[\text{Mean of This Summer} = \frac{150 + 170 + 150 + 190 + 180}{5} = 168.0\][/tex]
2. Calculate the variance for each summer:
- Last Summer:
- Variance measures the dispersion of the data points around the mean.
- [tex]\[\text{Variance of Last Summer} = 664.0\][/tex]
- This Summer:
- [tex]\[\text{Variance of This Summer} = 256.0\][/tex]
3. Calculate the standard deviation for each summer:
- Standard deviation is the square root of the variance and provides a measure of the spread of the data points.
- Last Summer:
- [tex]\[\text{Standard Deviation of Last Summer} = 25.77\][/tex]
- This Summer:
- [tex]\[\text{Standard Deviation of This Summer} = 16.0\][/tex]
4. Determine the inferences based on the calculations:
- Inference 1: "The books she read this summer had more pages, on average, than last summer."
- Comparing the means:
- [tex]\[\text{Mean This Summer} = 168.0\][/tex]
- [tex]\[\text{Mean Last Summer} = 184.0\][/tex]
- Since 168.0 is less than 184.0, this inference is False.
- Inference 2: "The number of pages in each book varied more last summer than this summer."
- Comparing the variances:
- [tex]\[\text{Variance Last Summer} = 664.0\][/tex]
- [tex]\[\text{Variance This Summer} = 256.0\][/tex]
- Since 664.0 is greater than 256.0, this inference is True.
- Inference 3: "Last summer, the average book had nearly twice the number of pages than the average book has this summer."
- To check this, see if the mean last summer is nearly twice the mean this summer:
- [tex]\[\text{Mean Last Summer} = 184.0\][/tex]
- [tex]\[\text{2 Mean This Summer} = 2 168.0 = 336.0\][/tex]
- Since 184.0 is not close to 336.0, this inference is False.
- Inference 4: "This summer, the number of pages in a book varied nearly twice as much as the number of pages in last summer's books."
- To check this, see if the variance this summer is nearly twice the variance last summer:
- [tex]\[\text{2 Variance Last Summer} = 2 664.0 = 1328.0\][/tex]
- [tex]\[\text{Variance This Summer} = 256.0\][/tex]
- Since 256.0 is not close to 1328.0, this inference is False.
Based on the step-by-step analysis, the correct inferences are:
1. The books she read this summer had more pages, on average, than last summer. (False)
2. The number of pages in each book varied more last summer than this summer. (True)
3. Last summer, the average book had nearly twice the number of pages than the average book has this summer. (False)
4. This summer, the number of pages in a book varied nearly twice as much as the number of pages in last summer's books. (False)
