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The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\overline{x}$ of the sample is very likely to get closer and closer to µ . This is discussed in more detail later in the text.
You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.
# of movies | Relative Frequency |
---|---|
0 | $$\frac{5}{30}$$ |
1 | $$\frac{15}{30}$$ |
2 | $$\frac{6}{30}$$ |
3 | $$\frac{3}{30}$$ |
4 | $$\frac{1}{30}$$ |
If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution .
A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean $\overline{x}$ is an example of a statistic which estimates the population mean μ .
When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: mean = $\frac{data\text{}sum}{number\text{}of\text{}data\text{}values}$ We simply need to modify the definition to fit within the restrictions of a frequency table.
Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is $\frac{lower\text{}boundary+upper\text{}boundary}{2}$ . We can now modify the mean definition to be $Mean\text{}of\text{}Frequency\text{}Table=\frac{{\displaystyle \sum fm}}{{\displaystyle \sum f}}$ where f = the frequency of the interval and m = the midpoint of the interval.
A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean.
Grade Interval | Number of Students |
---|---|
50–56.5 | 1 |
56.5–62.5 | 0 |
62.5–68.5 | 4 |
68.5–74.5 | 4 |
74.5–80.5 | 2 |
80.5–86.5 | 3 |
86.5–92.5 | 4 |
92.5–98.5 | 1 |
Grade Interval | Midpoint |
---|---|
50–56.5 | 53.25 |
56.5–62.5 | 59.5 |
62.5–68.5 | 65.5 |
68.5–74.5 | 71.5 |
74.5–80.5 | 77.5 |
80.5–86.5 | 83.5 |
86.5–92.5 | 89.5 |
92.5–98.5 | 95.5 |
Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:
Hours Teenagers Spend on Video Games | Number of Teenagers |
---|---|
0–3.5 | 3 |
3.5–7.5 | 7 |
7.5–11.5 | 12 |
11.5–15.5 | 7 |
15.5–19.5 | 9 |
What is the best estimate for the mean number of hours spent playing video games?
Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers
The midpoints are 1.75, 5.5, 9.5, 13.5,17.5.
Mean = (1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9) = 409.75
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