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Sagot :
Answer:
(a): The 95% confidence interval is (46.4, 53.6)
(b): The 95% confidence interval is (47.9, 52.1)
(c): Larger sample gives a smaller margin of error.
Step-by-step explanation:
Given
[tex]n = 30[/tex] -- sample size
[tex]\bar x = 50[/tex] -- sample mean
[tex]\sigma = 10[/tex] --- sample standard deviation
Solving (a): The confidence interval of the population mean
Calculate the standard error
[tex]\sigma_x = \frac{\sigma}{\sqrt n}[/tex]
[tex]\sigma_x = \frac{10}{\sqrt {30}}[/tex]
[tex]\sigma_x = \frac{10}{5.478}[/tex]
[tex]\sigma_x = 1.825[/tex]
The 95% confidence interval for the z value is:
[tex]z = 1.960[/tex]
Calculate margin of error (E)
[tex]E = z * \sigma_x[/tex]
[tex]E = 1.960 * 1.825[/tex]
[tex]E = 3.577[/tex]
The confidence bound is:
[tex]Lower = \bar x - E[/tex]
[tex]Lower = 50 - 3.577[/tex]
[tex]Lower = 46.423[/tex]
[tex]Lower = 46.4[/tex] --- approximated
[tex]Upper = \bar x + E[/tex]
[tex]Upper = 50 + 3.577[/tex]
[tex]Upper = 53.577[/tex]
[tex]Upper = 53.6[/tex] --- approximated
So, the 95% confidence interval is (46.4, 53.6)
Solving (b): The confidence interval of the population mean if mean = 90
First, calculate the standard error of the mean
[tex]\sigma_x = \frac{\sigma}{\sqrt n}[/tex]
[tex]\sigma_x = \frac{10}{\sqrt {90}}[/tex]
[tex]\sigma_x = \frac{10}{9.49}[/tex]
[tex]\sigma_x = 1.054[/tex]
The 95% confidence interval for the z value is:
[tex]z = 1.960[/tex]
Calculate margin of error (E)
[tex]E = z * \sigma_x[/tex]
[tex]E = 1.960 * 1.054[/tex]
[tex]E = 2.06584[/tex]
The confidence bound is:
[tex]Lower = \bar x - E[/tex]
[tex]Lower = 50 - 2.06584[/tex]
[tex]Lower = 47.93416[/tex]
[tex]Lower = 47.9[/tex] --- approximated
[tex]Upper = \bar x + E[/tex]
[tex]Upper = 50 + 2.06584[/tex]
[tex]Upper = 52.06584[/tex]
[tex]Upper = 52.1[/tex] --- approximated
So, the 95% confidence interval is (47.9, 52.1)
Solving (c): Effect of larger sample size on margin of error
In (a), we have:
[tex]n = 30[/tex] [tex]E = 3.577[/tex]
In (b), we have:
[tex]n = 90[/tex] [tex]E = 2.06584[/tex]
Notice that the margin of error decreases when the sample size increases.
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