Problem 1: Joshua wants to replicate a study where the lowest observed value is 12.4 while the highest is 12.8. He wants to estimate the population mean to within an error of 0.025 of its true value. Using 99% confidence level, find the sample size n that he needs. Hint: Since the range R= 12.8-12.4 = 0.4, then σ=R4=0.1
here are two things to remember when we decide on the quality of the
| Formula in Determining the Minimum Sample Size Needed when Estimating the Population Mean |
The computing formula is derived from the formula of the Margin of Error E=zα2(σn√)
n=(zα2∙σE)2
Note: Since the value of σσ is usually unknown, it can be estimated by the standard deviation s. We round up the obtained value to ensure that the sample size will be sufficient to achieve the specified reliability.
Consider the following illustration:
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Problem: In a certain village, Leony wants to estimate the mean weight μ, in kilograms, of all six-year old children to be included in a feeding program. She wants to be 99% confident that the estimate of μμis accurate to within 0.06 kg. Suppose from a previous study, the standard deviation of the weights of the target population was 0.5 kg., what should the sample size be? |
Solution:
Given the confidence level as 99%, we have, α=1−0.99=0.01.Hence, the confidence coefficient is 2.58.
The phrase " accurate to within 0.06 kg." indicates a narrowed width of the confidence interval. Thus, the desired margin of error E= 0.06 kg.
Population standard deviation σ=0.5kg.
Formula: n=(zα2∙σE)2
n=((2.58)(0.5)0.06)2=462.25,rounded up to 463
Therefore, the required sample size is 463 six-year old children.
| Formula in Determining the Minimum Sample Size Needed when Estimating the Population Proportion |
Note: In computing the sample size for estimating a population proportion, there are two situations to consider:
1. If some approximation of is known, that value may be used in the formula.
2. If no approximation is known, use =0.5.
This value will result in a sample size large enough to guarantee prediction given the confidence interval and the error of estimate.
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Steps in Solving for the Sample Size Involving Proportions Step 1: Determine the confidence level. Step 2: Determine the confidence coefficient. Step 3: Determine the margin of Error E. Step 4: Determine and . Step 5: Substitute the values in the formula, Step 6: Round up the resulting value to the nearest whole number. |
Illustration:
Problem:
Mary wants to know, with 95% confidence, the proportion of households who like to use detergent X. A previous survey showed that 42% like to use detergent X. Mary likes to be accurate within 2% of the true proportion. What sample size doe Mary need?
Solution:
| STEPS | SOLUTION |
| Step 1: Determine the confidence level. | 95% confidence |
| Step 2: Determine the confidence coefficient. | zα2=1.96 |
| Step 3: Determine the margin of Error E. | E= 2% = 0.02 |
| Step 4: Determine and . |
= 42% = 0.42 = 1 - 0.42 = 0.58 |
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Step 5: Substitute the values in the formula, Step 6: Round up the result. |
n=(0.42)(0.58)(1.960.02)2=2339.53≈2340 |
| Answer: | Mary needs a sample of 2340 respondents. |
Your TRY!!!
Directions: Determine the minimum sample size needed for the following:
Present your complete solution following the necessary steps and the formula.
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Problem 1: Joshua wants to replicate a study where the lowest observed value is 12.4 while the highest is 12.8. He wants to estimate the population mean to within an error of 0.025 of its true value. Using 99% confidence level, find the sample size n that he needs. Hint: Since the |
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Problem 2: The Dean of a college wants to use the proportion of a population to determine the sample size needed to interview reagrding their thoughts about the new school structure. He wants to be able to assert with a probability 0.95 that his error will be at most 0.05. Similar polls in the past showed that 65% approved the new structure. How large a sample does the Dean need? |
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