Question

Using the table for many repetitions of the experiment of tossing a coin 10 times, what are the mean number of heads in 50 repetitions of this experiment and 500 repetitions and which of these sample means is closer to the population mean? Number of Heads 0 1 2 3 4 5 6 7 8 9 10

In 50 Repetitions 0 0 1 5 9 16 10 6 2 1 0
In 200 Repetitions 0 2 8 22 38 55 41 24 9 1 0
In 500 Repetitions 2 5 24 57 111 111 110 56 21 3 0
In 1000 Repetitions 1 8 43 117 207 248 203 121 45 6 1

4.8, 5.03; 50 repetitions should be closer to the population mean
4.9, 4.99; 50 repetitions should be closer to the population mean
5.2, 4.94; 500 repetitions should be closer to the population mean
5.1, 4.9; 500 repetitions should be closer to the population mean

Answer 1

5.2, 4.94; 500 repetitions should be closer to the population mean

Explanation:

A. Mean of 50 repetitions

`\begin{array}{rrr}\\\mathbf{f} &\mathbf{x} &\mathbf{f\cdot x}\\0 & 0 & 0\\1 & 0 & 0\\2 & 1 & 2\\3 & 5 & 15\\4 & 9 & 36\\5 & 16 & 80\\6 & 10 & 60\\7 & 6 & 42\\8 & 2 & 16\\9 & 1 &9\\10 & 0 & 0\\\text{Sum =}&\mathbf{50} &\mathbf{260}\\\end{array}\\\\\text{Mean} = \frac{\sum{f \cdot x}}{n} = (260)/(50) = \mathbf{5.20}`

B. Mean of 500 repetitions

`\begin{array}{rrr}\\\mathbf{f} &\mathbf{x} &\mathbf{f\cdot x}\\0 & 2 & 0\\1 & 5 & 5\\2 & 24 & 48\\3 & 57 & 171\\4 & 111 & 444\\5 & 111 & 555\\6 & 110 & 660\\7 & 56 & 392\\8 & 21 & 168\\9 & 3 &27\\10 & 0 & 0\\\text{Sum =}&\mathbf{500} &\mathbf{2470}\\\end{array}\\\\\text{Mean} = \frac{\sum{f \cdot x}}{n} = (2470)/(500) = \mathbf{4.94}`

500 repetitions should be closer to the population mean because. Per the Central Limit Theorem, the mean of the sample should approach the mean of the population as the sample size increases.