From the given sample data in the table on the backpack weight, we have;
(a) The means of the samples are;
- First sample = 6.375
- Second sample = 6.375
- Third sample = 6.625
(b) The range is 0.25
(c) The true statements are;
- A single sample mean will tend to be a worse estimate than the mean of the sample means
- The farther the range of the sample means is from zero, the less confident they can be their estimate.
How can the mean of the sample means be found?
(a) The sample means of each of each of the three samples are found as follows;
[tex]Mean = \frac{ \sum \: x}{n} [/tex]
Where;
x = The value of a data point
[tex] \sum \: x = Sum of the data [/tex]
n = Sample size
The mean of the first sample, S1, data is therefore;
[tex]Mean \: S1 = \frac{3 + 7 + 8 + 3 + 7 + 9 + 6 + 8}{8} [/tex]
3+7+8+3+7+9+6+8 = 51
Which gives
[tex]Mean \: S1 = \frac{51}{8} = \mathbf{6.375\[/tex]
- Mean of the first sample = 6.375
Similarly, we have;
[tex]Mean \: S2 = \frac{8 + 6 + 4 + 7 + 9 + 4 + 6 + 7}{8} [/tex]
8 +6+4+7+9+4+6+7 = 51
Which gives;
[tex]Mean \: S2 = \mathbf{\frac{51}{8}} = 6.375 [/tex]
Mean of the second sample = 6.375
[tex]Mean \: S3 = \frac{9 + 4 + 5 + 8 + 7 + 5 + 9 + 6}{8} [/tex]
9+4+5+8+7+5+9+6 = 53
Which gives;
[tex]Mean \: S3 = \frac{53}{8} = \mathbf{6.625} [/tex]
- Mean of the third sample = 6.625
(b) The range of the means of the sample means is found as follows;
Range = Largest value - Smallest value
Which gives;
- Range of the sample means = 6.625 - 6.375 = 0.25
(c) The population mean is given by the mean of the sample means. That is, a very good estimate of the sample mean is given by the mean of the sample means.
The true statements are therefore;
- A single sample mean will tend to be a worse estimate than the mean of the sample means
- The farther the range of the sample means is from zero, the less confident they can be their estimate.
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