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9.5 Sample Size Considerations

Learning Objective

  1. To learn how to apply formulas for estimating the size samples that will be needed in order to construct a confidence interval for the difference in two population means or proportions that meets given criteria.

As was pointed out at the beginning of Section 7.4 "Sample Size Considerations" in Chapter 7 "Estimation", sampling is typically done with definite objectives in mind. For example, a physician might wish to estimate the difference in the average amount of sleep gotten by patients suffering a certain condition with the average amount of sleep got by healthy adults, at 90% confidence and to within half an hour. Since sampling costs time, effort, and money, it would be useful to be able to estimate the smallest size samples that are likely to meet these criteria.

Estimating μ1−μ2 with Independent Samples

Assuming that large samples will be required, the confidence interval formula for estimating the difference μ1−μ2 between two population means using independent samples is (ˉx1−ˉx2)±E, where

E=zα∕2√s21n1+s22n2

To say that we wish to estimate the mean to within a certain number of units means that we want the margin of error E to be no larger than that number. The number zα∕2 is determined by the desired level of confidence.

The numbers s1 and s2 are estimates of the standard deviations σ1 and σ2 of the two populations. In analogy with what we did in Section 7.4 "Sample Size Considerations" in Chapter 7 "Estimation" we will assume that we either know or can reasonably approximate σ1 and σ2.

We cannot solve for both n1 and n2, so we have to make an assumption about their relative sizes. We will specify that they be equal. With these assumptions we obtain the minimum sample sizes needed by solving the equation displayed just above for n1=n2.

Minimum Equal Sample Sizes for Estimating the Difference in the Means of Two Populations Using Independent Samples

The estimated minimum equal sample sizes n1=n2 needed to estimate the difference μ1−μ2 in two population means to within E units at 100(1−α)% confidence is

n1=n2=(zα∕2)2(σ21+σ22)E2(rounded up)

In all the examples and exercises the population standard deviations σ1 and σ2 will be given.

Example 13

A law firm wishes to estimate the difference in the mean delivery time of documents sent between two of its offices by two different courier companies, to within half an hour and with 99.5% confidence. From their records it will randomly sample the same number n of documents as delivered by each courier company. Determine how large n must be if the estimated standard deviations of the delivery times are 0.75 hour for one company and 1.15 hours for the other.

Solution:

Confidence level 99.5% means that α=1−0.995=0.005 so α∕2=0.0025. From the last line of Figure 12.3 "Critical Values of " we obtain z0.0025=2.807.

To say that the estimate is to be “to within half an hour” means that E = 0.5. Thus

n=(zα∕2)2(σ21+σ22)E2=(2.807)2(0.752+1.152)0.52=59.40953746

which we round up to 60, since it is impossible to take a fractional observation. The law firm must sample 60 document deliveries by each company.

Estimating μ1−μ2 with Paired Samples

As we mentioned at the end of Section 9.3 "Comparison of Two Population Means: Paired Samples", if the sample is large (meaning that n ≥ 30) then in the formula for the confidence interval we may replace tα∕2 by zα∕2, so that the confidence interval formula becomes ˉd±E for

E=zα∕2sd√n

The number sd is an estimate of the standard deviations σd of the population of differences. We must assume that we either know or can reasonably approximate σd. Thus, assuming that large samples will be required to meet the criteria given, we can solve the displayed equation for n to obtain an estimate of the number of pairs needed in the sample.

Minimum Sample Size for Estimating the Difference in the Means of Two Populations Using Paired Difference Samples

The estimated minimum number of pairs n needed to estimate the difference μd=μ1−μ2 in two population means to within E units at 100(1−α)% confidence using paired difference samples is

n=(zα∕2)2σ2dE2(rounded up)

In all the examples and exercises the population standard deviation of the differences σd will be given.

Example 14

A automotive tire manufacturer wishes to compare the mean lifetime of two tread designs under actual driving conditions. They will mount one of each type of tire on n vehicles (both on the front or both on the back) and measure the difference in remaining tread after 20,000 miles of driving. If the standard deviation of the differences is assumed to be 0.025 inch, find the minimum samples size needed to estimate the difference in mean depth (at 20,000 miles use) to within 0.01 inch at 99.9% confidence.

Solution:

Confidence level 99.9% means that α=1−0.999=0.001 so α∕2=0.0005. From the last line of Figure 12.3 "Critical Values of " we obtain z0.0005=3.291.

To say that the estimate is to be “to within 0.01 inch” means that E = 0.01. Thus

n=(zα∕2)2σ2dE2=(3.291)2(0.025)2(0.01)2=67.69175625

which we round up to 68. The manufacturer must test 68 pairs of tires.

Estimating p1−p2

The confidence interval formula for estimating the difference p1−p2 between two population proportions is ˆp1−ˆp2±E, where

E=zα∕2√ˆp1(1−ˆp1)n1+ˆp2(1−ˆp2)n2

To say that we wish to estimate the mean to within a certain number of units means that we want the margin of error E to be no larger than that number. The number zα∕2 is determined by the desired level of confidence.

We cannot solve for both n1 and n2, so we have to make an assumption about their relative sizes. We will specify that they be equal. With these assumptions we obtain the minimum sample sizes needed by solving the displayed equation for n1=n2.

Minimum Equal Sample Sizes for Estimating the Difference in Two Population Proportions

The estimated minimum equal sample sizes n1=n2 needed to estimate the difference p1−p2 in two population proportions to within E percentage points at 100(1−α)% confidence is

n1=n2=(zα∕2)2(ˆp1(1−ˆp1)+ˆp2(1−ˆp2))E2(rounded up)

Here we face the same dilemma that we encountered in the case of a single population proportion: the formula for estimating how large a sample to take contains the numbers ˆp1 and ˆp2, which we know only after we have taken the sample. There are two ways out of this dilemma. Typically the researcher will have some idea as to the values of the population proportions p1 and p2, hence of what the sample proportions ˆp1 and ˆp2 are likely to be. If so, those estimates can be used in the formula.

The second approach to resolving the dilemma is simply to replace each of ˆp1 and ˆp2 in the formula by 0.5. As in the one-population case, this is the most conservative estimate, since it gives the largest possible estimate of n. If we have an estimate of only one of p1 and p2 we can use that estimate for it, and use the conservative estimate 0.5 for the other.

Example 15

Find the minimum equal sample sizes necessary to construct a 98% confidence interval for the difference p1−p2 with a margin of error E = 0.05,

  1. assuming that no prior knowledge about p1 or p2 is available; and
  2. assuming that prior studies suggest that p1≈0.2 and p2≈0.3.

Solution:

Confidence level 98% means that α=1−0.98=0.02 so α∕2=0.01. From the last line of Figure 12.3 "Critical Values of " we obtain z0.01=2.326.

  1. Since there is no prior knowledge of p1 or p2 we make the most conservative estimate that ˆp1=0.5 and ˆp2=0.5. Then

    n1=n2=(zα∕2)2(ˆp1(1−ˆp1)+ˆp2(1−ˆp2))E2=(2.326)2((0.5)(0.5)+(0.5)(0.5))0.052=1082.0552

    which we round up to 1,083. We must take a sample of size 1,083 from each population.

  2. Since p1≈0.2 we estimate ˆp1 by 0.2, and since p2≈0.3 we estimate ˆp2 by 0.3. Thus we obtain

    n1=n2=(zα∕2)2(ˆp1(1−ˆp1)+ˆp2(1−ˆp2))E2=(2.326)2((0.2)(0.8)+(0.3)(0.7))0.052=800.720848

    which we round up to 801. We must take a sample of size 801 from each population.

Key Takeaways

  • If the population standard deviations σ1 and σ2 are known or can be estimated, then the minimum equal sizes of independent samples needed to obtain a confidence interval for the difference μ1−μ2 in two population means with a given maximum error of the estimate E and a given level of confidence can be estimated.
  • If the standard deviation σd of the population of differences in pairs drawn from two populations is known or can be estimated, then the minimum number of sample pairs needed under paired difference sampling to obtain a confidence interval for the difference μd=μ1−μ2 in two population means with a given maximum error of the estimate E and a given level of confidence can be estimated.
  • The minimum equal sample sizes needed to obtain a confidence interval for the difference in two population proportions with a given maximum error of the estimate and a given level of confidence can always be estimated. If there is prior knowledge of the population proportions p1 and p2 then the estimate can be sharpened.

Exercises

    Basic

  1. Estimate the common sample size n of equally sized independent samples needed to estimate μ1−μ2 as specified when the population standard deviations are as shown.

    1. 90% confidence, to within 3 units, σ1=10 and σ2=7
    2. 99% confidence, to within 4 units, σ1=6.8 and σ2=9.3
    3. 95% confidence, to within 5 units, σ1=22.6 and σ2=31.8

  2. Estimate the common sample size n of equally sized independent samples needed to estimate μ1−μ2 as specified when the population standard deviations are as shown.

    1. 80% confidence, to within 2 units, σ1=14 and σ2=23
    2. 90% confidence, to within 0.3 units, σ1=1.3 and σ2=0.8
    3. 99% confidence, to within 11 units, σ1=42 and σ2=37

  3. Estimate the number n of pairs that must be sampled in order to estimate μd=μ1−μ2 as specified when the standard deviation sd of the population of differences is as shown.

    1. 80% confidence, to within 6 units, σd=26.5
    2. 95% confidence, to within 4 units, σd=12
    3. 90% confidence, to within 5.2 units, σd=11.3

  4. Estimate the number n of pairs that must be sampled in order to estimate μd=μ1−μ2 as specified when the standard deviation sd of the population of differences is as shown.

    1. 90% confidence, to within 20 units, σd=75.5
    2. 95% confidence, to within 11 units, σd=31.4
    3. 99% confidence, to within 1.8 units, σd=4

  5. Estimate the minimum equal sample sizes n1=n2 necessary in order to estimate p1−p2 as specified.

    1. 80% confidence, to within 0.05 (five percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.20 and p2≈0.65

    2. 90% confidence, to within 0.02 (two percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.75 and p2≈0.63

    3. 95% confidence, to within 0.10 (ten percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.11 and p2≈0.37

  6. Estimate the minimum equal sample sizes n1=n2 necessary in order to estimate p1−p2 as specified.

    1. 80% confidence, to within 0.02 (two percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.78 and p2≈0.65

    2. 90% confidence, to within 0.05 (two percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.12 and p2≈0.24

    3. 95% confidence, to within 0.10 (ten percentage points)

      1. when no prior knowledge of p1 or p2 is available
      2. when prior studies indicate that p1≈0.14 and p2≈0.21

    Applications

  1. An educational researcher wishes to estimate the difference in average scores of elementary school children on two versions of a 100-point standardized test, at 99% confidence and to within two points. Estimate the minimum equal sample sizes necessary if it is known that the standard deviation of scores on different versions of such tests is 4.9.

  2. A university administrator wishes to estimate the difference in mean grade point averages among all men affiliated with fraternities and all unaffiliated men, with 95% confidence and to within 0.15. It is known from prior studies that the standard deviations of grade point averages in the two groups have common value 0.4. Estimate the minimum equal sample sizes necessary to meet these criteria.

  3. An automotive tire manufacturer wishes to estimate the difference in mean wear of tires manufactured with an experimental material and ordinary production tire, with 90% confidence and to within 0.5 mm. To eliminate extraneous factors arising from different driving conditions the tires will be tested in pairs on the same vehicles. It is known from prior studies that the standard deviations of the differences of wear of tires constructed with the two kinds of materials is 1.75 mm. Estimate the minimum number of pairs in the sample necessary to meet these criteria.

  4. To assess to the relative happiness of men and women in their marriages, a marriage counselor plans to administer a test measuring happiness in marriage to n randomly selected married couples, record the their test scores, find the differences, and then draw inferences on the possible difference. Let μ1 and μ2 be the true average levels of happiness in marriage for men and women respectively as measured by this test. Suppose it is desired to find a 90% confidence interval for estimating μd=μ1−μ2 to within two test points. Suppose further that, from prior studies, it is known that the standard deviation of the differences in test scores is σd≈10. What is the minimum number of married couples that must be included in this study?

  5. A journalist plans to interview an equal number of members of two political parties to compare the proportions in each party who favor a proposal to allow citizens with a proper license to carry a concealed handgun in public parks. Let p1 and p2 be the true proportions of members of the two parties who are in favor of the proposal. Suppose it is desired to find a 95% confidence interval for estimating p1−p2 to within 0.05. Estimate the minimum equal number of members of each party that must be sampled to meet these criteria.

  6. A member of the state board of education wants to compare the proportions of National Board Certified (NBC) teachers in private high schools and in public high schools in the state. His study plan calls for an equal number of private school teachers and public school teachers to be included in the study. Let p1 and p2 be these proportions. Suppose it is desired to find a 99% confidence interval that estimates p1−p2 to within 0.05.

    1. Supposing that both proportions are known, from a prior study, to be approximately 0.15, compute the minimum common sample size needed.
    2. Compute the minimum common sample size needed on the supposition that nothing is known about the values of p1 and p2.

Answers

    1. n1=n2=45,
    2. n1=n2=56.
    3. n1=n2=234
    1. n1=n2=33.
    2. n1=n2=35.
    3. n1=n2=13
      1. n1=n2=329,
      2. n1=n2=255.
      1. n1=n2=3383,
      2. n1=n2=2846.
      1. n1=n2=193,
      2. n1=n2=128

  1. n1=n2≈80

  3. n1=n2≈34

  5. n1=n2≈769