In a preliminary study, the sample standard deviation for the duration of a particular
back pain suffered by patients was 18.0 months. How large a random sample is
needed to construct a 90% confidence interval so that an estimate can be made within
2 months of the actual duratio
Formula:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be 1.645 using a z-table to represent the 90% confidence interval, sd = 18, E = 2, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
To determine the sample size needed to construct a 90% confidence interval within 2 months of the actual duration, we can use the following formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (90% confidence level corresponds to a Z-score of approximately 1.645)
σ = standard deviation
E = margin of error
In this case, the standard deviation (σ) is given as 18.0 months, and the margin of error (E) is 2 months. So, plugging in the values, the formula becomes:
n = (1.645 * 18.0 / 2)^2
Calculating the equation:
n = (1.645 * 18.0 / 2)^2
n = 15.051225
Therefore, to construct a 90% confidence interval within 2 months of the actual duration, a sample size of approximately 16 patients is needed.
To determine the sample size needed to construct a confidence interval, we need to use the formula:
n = (Z * σ / E)²
where:
n = sample size
Z = Z-score corresponding to the desired confidence level
σ = standard deviation of the population
E = maximum allowable error (margin of error)
In this case, we want to construct a 90% confidence interval with a maximum allowable error of 2 months.
Step 1: Find the Z-score for a 90% confidence level
We can use a Z-table or a calculator to find the Z-score that corresponds to a 90% confidence level. The Z-score for a 90% confidence level is approximately 1.645.
Step 2: Plug the values into the formula
n = (Z * σ / E)²
n = (1.645 * 18.0 / 2)²
n = (29.61 / 2)²
n = 14.805²
n ≈ 219.12
Step 3: Round up to the nearest whole number
Since we can't have a fraction of a sample, we round up the calculated sample size to the nearest whole number. Therefore, the required sample size is approximately 220.
So, to construct a 90% confidence interval with an estimate within 2 months of the actual duration, you would need a random sample of at least 220 patients.