08.03)Esther wants to know if the number of words on a page in her language arts book is generally more than the number of words on a page in her social studies book. She takes a random sample of 25 pages in each book and then calculates the mean, median, and mean absolute deviation for the 25 samples of each book.
Book Mean Median Mean Absolute Deviation
Language arts 78.5 60 14.2
Social studies 68.7 65 9.8
She claims that because the mean number of words on each page of the language arts book is greater than the mean number of words on each page of the social studies book, the language arts book has more words per page. Based on the data, is this a valid inference?
because there is too much variability in the data
anyway hope that helps have a good day :)
Based on the data, this is not a valid inference.
I assume that "absolution deviation" means the same as standard deviation (SD).
Z = (score-mean)/SD
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
related to the Z score to compare with the level of significance you are using.
To determine if Esther's inference is valid based on the given data, let's analyze the information provided:
The mean (average) measures the central tendency of a dataset, the median represents the middle value in an ordered set of data, and the mean absolute deviation measures how spread out the data points are from the mean.
According to the data:
For the Language Arts book:
- Mean: 78.5 words per page
- Median: 60 words per page
- Mean Absolute Deviation: 14.2
For the Social Studies book:
- Mean: 68.7 words per page
- Median: 65 words per page
- Mean Absolute Deviation: 9.8
While the mean number of words per page in the Language Arts book (78.5) is greater than the mean number of words per page in the Social Studies book (68.7), this does not necessarily imply that the Language Arts book has more words per page.
Here's why:
1. Median: The median is a better measure of central tendency when dealing with skewed datasets. In this case, the median for the Language Arts book is lower (60) than the median for the Social Studies book (65). This suggests that at least half of the Language Arts book pages have fewer words than the median for the Social Studies book.
2. Mean Absolute Deviation: The mean absolute deviation indicates the average amount by which the data points deviate from the mean. A higher mean absolute deviation suggests more variability in the data. In this case, the mean absolute deviation for the Language Arts book (14.2) is greater than that of the Social Studies book (9.8). This implies that the word count on each page in the Language Arts book tends to deviate more from the mean compared to the Social Studies book.
Based on these two additional measures, it is not valid to infer that the language arts book has more words per page just because its mean is higher. The median suggests that it may have a lower word count on most pages, and the larger mean absolute deviation implies more variability in word counts.
Therefore, it is crucial to consider multiple measures, such as median and mean absolute deviation, in addition to the mean, to make a valid inference about the number of words per page in each book.