A federal funding program is available to low-income neighborhoods. To qualify for funding, a neighborhood must have a mean household income of less than $15,000 per year. Neighborhoods with mean annual household income of $15,000 or more do not qualify. Funding decisions are based on a sample of residents in the neighborhood. A hypothesis test with a .02 level of significance is conducted. If the funding guidelines call for a maximum probability of .05 of not funding a neighborhood with a mean annual household income of $14,000, what sample size should be used in the funding decision study? Use = $4000 as a planning value.
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To determine the sample size needed for the funding decision study, we can use the formula for sample size estimation for a hypothesis test:
n = (Z * σ / E)^2
Where:
n = sample size
Z = critical value for the desired level of significance (in this case, 0.02)
σ = standard deviation of the population (we'll use $4000 as the planning value)
E = margin of error (the maximum probability of not funding a neighborhood)
First, let's find the critical value Z for a significance level of 0.02. Since it is a two-tailed test, we need to split the significance level between two tails:
Significance level for one tail = 0.02 / 2 = 0.01
Using a standard normal distribution table or calculator, we can find the Z-score corresponding to a cumulative probability of 0.01. Let's denote it as Zc.
Next, let's calculate the margin of error (E) using the maximum probability of not funding a neighborhood with a mean annual household income of $14,000, which is 0.05.
E = Zc * (σ / √n)
Now, we can rearrange the formula to solve for n:
n = (Z * σ / E)^2
n = [(Zc * σ) / E]^2
Let's plug in the given values:
Zc = the Z-score corresponding to a cumulative probability of 0.01
σ = $4000
E = 0.05
Calculate the critical Z-value Zc:
From the standard normal distribution table or calculator, the Z-score for a cumulative probability of 0.01 is approximately -2.33 (round to two decimal places).
Zc = -2.33
Calculate the sample size:
n = [(Zc * σ) / E]^2
n = [(-2.33 * $4000) / 0.05]^2
n = [(-9320) / 0.05]^2
n = [186,400]^2
n ≈ 34,765,600
Now, since we can't have a fraction of a person in a sample, we need to round up to the nearest whole number:
n ≈ 34,765,600
Therefore, a sample size of approximately 34,765,600 should be used in the funding decision study.
To determine the sample size required for the funding decision study, we need to consider the desired level of significance, the planning value, and the guidelines for funding.
Here are the steps to calculate the required sample size:
Step 1: Determine the level of significance (α)
In this case, the level of significance is given as 0.02, which means we want to limit the probability of not funding a neighborhood with a mean income of $14,000 to no more than 0.02.
Step 2: Determine the maximum probability of not funding a neighborhood (β)
The guidelines state that the maximum probability of not funding a neighborhood with a mean income of $14,000 should be no more than 0.05.
Step 3: Determine the difference between the planning value and the mean income to qualify for funding
The planning value is given as $4000, and the mean income to qualify for funding is $15,000. Therefore, the difference is $15,000 - $4000 = $11,000.
Step 4: Calculate the standard deviation (σ)
Since the standard deviation is not provided, we'll assume it to be unknown. In this case, we can estimate it using the planning value ($4000) as a planning value.
Step 5: Use a sample size formula for a one-sample test
Formula: n = (Zα + Zβ)² * (σ²) / (Δμ)²
Where:
- n is the required sample size
- Zα is the z-value for the desired level of significance (α)
- Zβ is the z-value for the maximum probability of not funding a neighborhood (β)
- σ is the standard deviation
- Δμ is the difference between the planning value and the mean income to qualify for funding
Step 6: Calculate the required sample size
Plug in the values into the formula and calculate accordingly:
n = (Zα + Zβ)² * (σ²) / (Δμ)²
Using a z-table or calculator, find the z-values for α = 0.02 and β = 0.05. Let's say the resulting z-values are Zα = 2.05 and Zβ = 1.645.
Substituting the values into the formula:
n = (2.05 + 1.645)² * (σ²) / (11,000)²
Now we need to know the value of the standard deviation (σ), which is not provided. Without this information, it is not possible to calculate the exact required sample size.
Therefore, to determine the specific sample size required in this funding decision study, we would need the standard deviation (σ) or additional information about the population.