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Suppose a random sample of size 44 is selected from a population with [tex]$\sigma = 11$[/tex]. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).

a. The population size is infinite (to 2 decimals).
Answer: 1.66

b. The population size is [tex]$N=50,000$[/tex] (to 2 decimals).
Answer: 1.66

c. The population size is [tex]$N=5000$[/tex] (to 2 decimals).

d. The population size is [tex]$N=500$[/tex] (to 2 decimals).
Answer: 1.59

Sagot :

GinnyAnsweridn
Certainly, let's go step by step to find the standard error of the mean in the given cases:

### Given Data:
- Sample size, [tex]\( n = 44 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 11 \)[/tex]

### Case a: The Population Size is Infinite
For an infinite population, the finite population correction factor is not needed. The standard error of the mean is calculated using the formula:

[tex]\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} \][/tex]

Substituting the given values:

[tex]\[ \text{Standard Error} = \frac{11}{\sqrt{44}} \][/tex]

Rounded to 2 decimals, the standard error is:

[tex]\[ \boxed{1.66} \][/tex]

### Case b: The Population Size is [tex]\( N = 50,000 \)[/tex]
When the population size is very large (50,000 in this case), it approximates an infinite population, so the finite population correction factor is essentially 1. The standard error remains:

[tex]\[ \text{Standard Error} = \frac{11}{\sqrt{44}} \][/tex]

Rounded to 2 decimals, the standard error is:

[tex]\[ \boxed{1.66} \][/tex]

### Case c: The Population Size is [tex]\( N = 5000 \)[/tex]
For a finite population, we need to apply the finite population correction factor. The formula for the finite population correction factor (FPC) is:

[tex]\[ \text{FPC} = \sqrt{\frac{N - n}{N - 1}} \][/tex]

Substituting [tex]\( N = 5000 \)[/tex]:

[tex]\[ \text{FPC} = \sqrt{\frac{5000 - 44}{5000 - 1}} \][/tex]

The standard error with the finite population correction is:

[tex]\[ \text{Standard Error} = \left( \frac{11}{\sqrt{44}} \right) \times \text{FPC} \][/tex]

[tex]\[ \boxed{1.65} \][/tex]

### Case d: The Population Size is [tex]\( N = 500 \)[/tex]
Similarly, for [tex]\( N = 500 \)[/tex]:

[tex]\[ \text{FPC} = \sqrt{\frac{500 - 44}{500 - 1}} \][/tex]

The standard error with the finite population correction is:

[tex]\[ \text{Standard Error} = \left( \frac{11}{\sqrt{44}} \right) \times \text{FPC} \][/tex]

Rounded to 2 decimals, the standard error is:

[tex]\[ \boxed{1.59} \][/tex]

So the final answers are:
- For an infinite population: [tex]\( \boxed{1.66} \)[/tex]
- For [tex]\( N = 50,000 \)[/tex]: [tex]\( \boxed{1.66} \)[/tex]
- For [tex]\( N = 5000 \)[/tex]: [tex]\( \boxed{1.65} \)[/tex]
- For [tex]\( N = 500 \)[/tex]: [tex]\( \boxed{1.59} \)[/tex]
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