A town has 5000 residents. A survey finds that 81 residents
out of a random sample of 625 residents have red hair. The margin of
error for the survey is 4%. Find the interval in which the total number
of residents with red hair is most likely to lie.

The range from 63 to 99 is where the overall population of people with red hair is most likely to fall in the interval.  

To solve:

The margin of error is equal to z (p (p - 1) / n).

4% is the error margin, We may therefore change the values to:

0.04 = z ×√ (1 - 81/625) / 625)

Then, using a standard normal table, we determine the z-value for 95%, which is around 1.96.

0.04 = 1.96 × √(81/625) × (1 - 81/625) / 625)

When we solve for (1 - 81/625)/625", we get a value of around 0.0098.

Calculations for the range are now being achieved: 

81 - (0.0098 5000) = 63 

81 + (0.0098 5000) = 99

Therefore, the range of residents with red hair that contains the greatest likelihood of lying is from 63 to 99.

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The interval in which the total number of residents with red hair is most likely to lie is approximately between 545 and 745.

To find the interval in which the total number of residents with red hair is most likely to lie, we can calculate the confidence interval based on the survey results and the margin of error.

The margin of error is given as 4%, which means we have a 96% confidence level (100% - 4%). This corresponds to a z-score of approximately 1.96 for a standard normal distribution.

First, we calculate the standard error using the formula:

Standard Error = sqrt((p * (1 - p)) / n),

where p is the proportion of red-haired residents in the sample (81/625) and n is the sample size (625).

Standard Error = sqrt((81/625) * (1 - 81/625) / 625) ≈ 0.0179.

Next, we can calculate the margin of error by multiplying the standard error by the z-score:

Margin of Error = 1.96 * 0.0179 ≈ 0.035.

Now, we can calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error, respectively:

Lower bound = (81/625) - 0.035 ≈ 0.109.

Upper bound = (81/625) + 0.035 ≈ 0.149.

To find the range or interval, we can multiply the lower and upper bounds by the total number of residents (5000):

Lower bound (rounded) = 0.109 * 5000 ≈ 545.

Upper bound (rounded) = 0.149 * 5000 ≈ 745.

Therefore, the interval in which the total number of residents with red hair is most likely to lie is approximately between 545 and 745.

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