10.6 - Compare a Single Mean

Example Section

Suppose the male population of an area in a developing country is known to have had a mean serum total cholesterol of 5.5 mmol/l 10 years ago, with an estimated standard deviation of 1.4 mmol/l. In recent years Western food has been imported into the country and is believed to have increased cholesterol levels. The investigators want to see if mean cholesterol levels have increased a clinically meaningful amount (up to about 6 mmol/l, a difference of 0.5 mmol/l) with a one-sided alpha of 0.05 and a power of 90%.

Formula Section

We are interested in testing the following hypothesis:

\(\mathrm{H}_{0}\colon \mu=\mu_{0}\)

\(\mathrm{H}_{1}\colon \mu=\mu_{1}\)

The formula needed to calculate the sample size is:

\(\displaystyle{n=\frac{\left(z_{\alpha}+z_{\beta}\right)^{2} \sigma^{2}}{\left(\mu_{1}-\mu_{0}\right)^{2}}}\)

Where

  • \(\mu_{0}\)= null hypothesized value
  • \(\mu_{1}\)= alternative hypothesized value
  • \(\sigma\) = standard deviation

From the formula, we can calculate that n=67.1, so rounding to the next whole number would be n=68 .

To use the table below, we can calculate S= (6.0 – 5.5)/1.4 = 0.3571. This exact value does not appear in Table B.7. In these situations, we can get a rough idea of sample size by taking the nearest figure for S. In the example, the nearest tabulated figure is 0.35, which has n = 70 (for one-sided 5% significance and 90% power). This is only slightly above the true value of 67 for S = 0.3571. However, this process can lead to considerable error when S is small, so it is preferable to use the formula. 

Sample Size Statement:  A total sample size of n=68 is needed to detect a 0.5 mmol/l increase in mean cholesterol compared to an historical value of 5.5 mmol/l using a one-group t- test with one-sided alpha of 0.05 and 90% power, and assuming a standard deviation of 1.4.  
 

Table B.7. Sample size requirements for testing the value of a single mean or the difference between two means.

The table gives requirements for testing a single mean with a one-sided test directly. For two-sided tests, use the column corresponding to half the required significance level. For tests of the difference between two means, the total sample size (for the two groups combined) is obtained by multiplying the requirement given below by 4 if the two samples sizes are equal or by \((r+1)^{2} / r\) if the ratio of the first to the second is \(r : 1\) (assuming equal variances).
Note that \(S\) = difference/standard deviation.
  5%
Significance		 2.5%
Significance		 1%
Significance		 0.5%
Significance		 0.1%
Significance		 0.05%
Significance
\(S\) 90%
Power		 95%
Power		 90%
Power		 95%
Power		 90%
Power		 95%
Power		 90%
Power		 95%
Power		 90%
Power		 95%
Power		 90%
Power		 95%
Power
0.01 85 639 108 222 105 075 129 948 130 170 157 705 148 794 178 142 191 125 224 211 209 040 243 580
0.02 21 410 27 056 26 269 32 487 32 543 39 427 37 199 44 536 47 782 56 053 52 260 60 895
0.03 9 516 12 025 11 675 14 439 14 464 17 523 16 533 19 794 21 237 24 913 23 227 27 065
0.04 5 353 6 764 6 568 8 122 8 136 9 587 9 300 11 134 11 946 14 014 13 065 15 224
0.05 3 426 4 329 4 203 5 198 5 207 6 309 5 952 7 126 7 645 8 969 8 362 9 744
0.06 2 379 3 007 2 919 3 610 3 616 4 381 4 134 4 949 5 310 6 229 5 807 6 767
0.07 1 748 2 209 2 145 2 652 2 657 3 219 3 037 3 636 3 901 4 576 4 267 4 972
0.08 1 339 1 691 1 642 2 031 2 034 2 465 2 325 2 784 2 987 3 504 3 267 3 806
0.09 1 058 1 334 1 298 1 605 1 608 1 947 1 837 2 200 2 360 2 769 2 581 3 008
0.10 857 1 083 1 051 1 300 1 302 1 578 1 488 1 782 1 912 2 243 2 091 2 436
0.15 381 481 467 578 579 701 662 792 850 997 930 1 083
0.20 215 271 263 325 326 395 372 446 478 561 523 609
0.25 138 174 169 208 209 253 239 286 306 359 335 390
0.30 96 121 117 145 145 176 166 198 213 250 233 271
0.35 70 89 86 107 107 129 122 146 157 184 171 199
0.40 54 68 66 82 82 99 93 112 120 141 131 153
0.45 43 54 52 65 65 78 74 88 95 111 104 121
0.50 35 44 43 52 53 64 60 72 77 90 84 98
0.55 29 36 35 43 44 53 50 59 64 75 70 81

from Woodward, M. Epidemiology Study Design and Analysis. Boca Raton: Chapman and Hall, 2013, p.770