i have no idea!
The real estate assessor for a country government wants to study various characteristics of single family houses in the country. A random sample of 70 houses reveals the following: the mean is 1759, standard deviation is 380 and 42 houses have central air conditioning.
How would I construct a 99% confidence interval estimate of the population mean heated area of the houses?
And the construct a 95% confidence interval estimate of the population proportion of houses that have central air conditioning.
Formula for 99% interval estimate of the population mean:
CI99 = mean + or - 2.575(sd/√n)
...where + or - 2.575 represents the 99% confidence interval using a z-table, sd = standard deviation, √ = square root, and n = sample size.
Substitute what you know into the formula:
CI99 = 1759 + or - 2.575(380/√70)
Finish the calculation.
Formula for 95% interval estimate of the population proportion:
CI95 = p + or - 1.96[√(pq/n]
...where + or - 1.96 represents the 95% confidence interval using a z-table, p = x/n, q = 1 - p, and n = sample size.
Substitute what you know into the formula:
CI95 = 42/70 + or - 1.96[√(42/70)(28/70)/70]
Convert the fractions to decimals and finish the calculation.
I hope this will help.
80008.3-79774.54
To construct a 99% confidence interval estimate of the population mean heated area of the houses, you can follow these steps:
Step 1: Define the required confidence level (99%), which determines the critical value. The critical value for a 99% confidence level can be found by subtracting the confidence level from 1 and dividing by 2. In this case, (1 - 0.99) / 2 = 0.005.
Step 2: Determine the critical value by finding the z-score associated with the desired level of confidence. You can use a standard normal distribution table or a statistical software to find the critical value. In this case, the critical value is approximately 2.58.
Step 3: Calculate the margin of error (E) using the formula: E = z * (standard deviation / sqrt(n)), where z is the critical value, standard deviation is the sample standard deviation, and n is the sample size. In this case, the standard deviation is 380 and the sample size is 70.
E = 2.58 * (380 / sqrt(70)) ≈ 109.79.
Step 4: Calculate the lower and upper bounds of the confidence interval using the formula: Lower bound = mean - E and Upper bound = mean + E. In this case, the mean is 1759.
Lower bound = 1759 - 109.79 ≈ 1649.21.
Upper bound = 1759 + 109.79 ≈ 1868.79.
Therefore, the 99% confidence interval estimate for the population mean heated area of the houses is approximately (1649.21, 1868.79).
To construct a 95% confidence interval estimate of the population proportion of houses that have central air conditioning, you can follow these steps:
Step 1: Define the required confidence level (95%), which determines the critical value. The critical value for a 95% confidence level can be found by subtracting the confidence level from 1 and dividing by 2. In this case, (1 - 0.95) / 2 = 0.025.
Step 2: Determine the critical value by finding the z-score associated with the desired level of confidence. You can use a standard normal distribution table or a statistical software to find the critical value. In this case, the critical value is approximately 1.96.
Step 3: Calculate the margin of error (E) using the formula: E = z * sqrt((p̂ * (1 - p̂)) / n), where z is the critical value, p̂ is the sample proportion, and n is the sample size. In this case, the sample proportion is 42/70.
p̂ = 42/70 ≈ 0.6.
E = 1.96 * sqrt((0.6 * (1 - 0.6)) / 70) ≈ 0.0985.
Step 4: Calculate the lower and upper bounds of the confidence interval using the formula: Lower bound = p̂ - E and Upper bound = p̂ + E. In this case, p̂ ≈ 0.6.
Lower bound = 0.6 - 0.0985 ≈ 0.5015.
Upper bound = 0.6 + 0.0985 ≈ 0.6985.
Therefore, the 95% confidence interval estimate for the population proportion of houses that have central air conditioning is approximately (0.5015, 0.6985).
To construct a 99% confidence interval estimate of the population mean heated area of the houses, you can follow these steps:
1. Identify the sample mean, sample standard deviation, and sample size:
- Sample mean (x̄): 1759
- Sample standard deviation (s): 380
- Sample size (n): 70
2. Calculate the standard error of the mean (SE):
- SE = s / √n = 380 / √70
3. Determine the critical value corresponding to a 99% confidence level. Since the sample size is large (n > 30), you can use a z-distribution. The critical value for a 99% confidence level is found by subtracting 1 minus the confidence level from 1, dividing by 2, and looking up the corresponding z-score using a Z-table or calculator. In this case, the critical value is approximately 2.57.
4. Calculate the margin of error (ME):
- ME = z * SE = 2.57 * (380 / √70)
5. Construct the confidence interval:
- Lower Limit (LL) = x̄ - ME
- Upper Limit (UL) = x̄ + ME
Therefore, the 99% confidence interval estimate of the population mean heated area of the houses would be (LL, UL), where LL and UL are the lower and upper limits of the interval obtained from the calculations above.
To construct a 95% confidence interval estimate of the population proportion of houses that have central air conditioning, you can follow these steps:
1. Identify the number of houses with central air conditioning and the sample size:
- Number of houses with central air conditioning (x): 42
- Sample size (n): 70
2. Calculate the sample proportion (p̂):
- p̂ = x / n = 42 / 70
3. Determine the critical value corresponding to a 95% confidence level. Since the sample size (n) is large (n > 30), you can use a z-distribution. The critical value for a 95% confidence level is found by subtracting 1 minus the confidence level from 1, dividing by 2, and looking up the corresponding z-score using a Z-table or calculator. In this case, the critical value is approximately 1.96.
4. Calculate the margin of error (ME):
- ME = z * √((p̂(1-p̂))/n) = 1.96 * √((42/70)(1-42/70))
5. Construct the confidence interval:
- Lower Limit (LL) = p̂ - ME
- Upper Limit (UL) = p̂ + ME
Therefore, the 95% confidence interval estimate of the population proportion of houses that have central air conditioning would be (LL, UL), where LL and UL are the lower and upper limits of the interval obtained from the calculations above.