You are a travel agent and wish to estimate, with 98% confidence, the proportion of vacationers who use an online service or the Internet to make travel reservations. Your estimate must be accurate within 4% of the population proportion.
a) No preliminary estimate is available. Find the minimum sample size needed.
b) Find the minimum sample size needed, using a prior study that found 30% of the respondents said they used an online service or the Internet to make travel reservations.
c) Compare the results form parts (a) and (b).
Show steps please, I really need help I don't understand how to set this up or solve it at all!
To estimate the minimum sample size needed in each case, we need to use the formula for sample size calculation for proportions:
n = (Z^2 * p * (1-p)) / (E^2)
Where:
- n is the required sample size.
- Z is the z-score corresponding to the desired level of confidence (98% confidence corresponds to a z-score of approximately 2.33).
- p is the estimated proportion of vacationers who use an online service or the Internet to make travel reservations.
- E is the desired margin of error.
a) When no preliminary estimate is available:
Since we don't have an initial estimate, we need to use a conservative estimate for p, which is 0.5 (maximum variance). The desired margin of error (E) is 4% (0.04).
Using the formula, we can calculate the minimum sample size needed:
n = (2.33^2 * 0.5 * (1-0.5)) / (0.04^2)
n = 539.54
So, the minimum sample size needed without a preliminary estimate is 540 (rounded up to the nearest whole number).
b) Using a prior study estimate:
In this case, we have a prior estimate of p, which is 30% (0.30). The desired margin of error (E) is still 4% (0.04). We can use the same formula:
n = (2.33^2 * 0.3 * (1-0.3)) / (0.04^2)
n = 478.30
So, the minimum sample size needed with a prior study estimate is 479 (rounded up to the nearest whole number).
c) Comparing the results:
When comparing the results from parts (a) and (b), we can see that having a prior estimate of p (part b) reduces the required sample size compared to not having a preliminary estimate (part a). This reduction in sample size is due to having some information about the population proportion already available.
In this case, using a prior estimate (part b) yields a smaller sample size requirement, which can save time and resources. However, if the prior estimate is not reliable or if there have been significant changes in the population, it may be better to use the conservative estimate (part a) to ensure a more accurate estimation.
To determine the minimum sample size needed, we can use the formula for sample size estimation in a proportion estimation problem:
n = (Z^2 * p * q) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (98% confidence corresponds to Z = 2.33)
p = estimated population proportion (usually a preliminary estimate is used)
q = 1 - p
E = maximum error (4% of the population proportion)
a) Without a preliminary estimate:
n = (2.33^2 * 0.5 * 0.5) / (0.04^2)
n = (5.4289 * 0.25) / 0.0016
n = 13.57225 / 0.0016
n ≈ 8483.9
For a 98% confidence level and an accuracy of within 4% of the population proportion, the minimum sample size needed is approximately 8484. Since you cannot have a fraction of a person, the next whole number, n, would be 8484.
b) With a preliminary estimate of 30%:
n = (2.33^2 * 0.3 * 0.7) / (0.04^2)
n = (5.4289 * 0.21) / 0.0016
n = 1.199909 / 0.0016
n ≈ 749.9434
For a 98% confidence level and an accuracy of within 4% of the population proportion, using the preliminary estimate of 30% from a prior study, the minimum sample size needed is approximately 750. Again, since you cannot have a fraction of a person, the next whole number, n, would be 750.
c) Comparing the results:
In part (a) without a preliminary estimate, the minimum sample size needed was approximately 8484. In part (b) with a preliminary estimate of 30%, the minimum sample size needed was approximately 750.
The sample size needed when a preliminary estimate is available (part b) is significantly smaller than when no preliminary estimate is available (part a). By having a preliminary estimate, we have some information about the population proportion, which allows for a smaller sample size to achieve the desired accuracy level.
Try this formula:
n = [(z-value)^2 * p * q]/E^2
= [(2.33)^2 * .5 * .5]/.04^2
= ? (round to the next highest whole number)
I'll let you finish the calculation.
Note: n = sample size needed; .5 for p and .5 for q are used if no value is stated in the problem. E = maximum error, which is .04 (4%) in the problem. Z-value is found using a z-table (for 98%, the value is 2.33).
Redo the problem using .3 for p and .7 for q. (Note: q = 1 - p)