Промышленный лизинг Промышленный лизинг  Методички 

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formula is SQRT(5% * 95% / 100,000) is about 0.07 percent. So, we are 68 percent confident that the actual response is between 4.93 percent and 5.07 percent. We have also observed that a bit over 95 percent is within two standard deviations; so the range of 4.86 percent and 5.14 percent is just over 95 percent confident. So, if we observe a 5 percent response rate for the challenger offer, then we are over 95 percent confident that the response rate on the whole population would have been between 4.86 percent and 5.14 percent. Note that this conclusion depends very much on the fact that people who got the challenger offer were selected randomly from the entire population.

Comparing Results Using Confidence Bounds

The previous section discussed confidence intervals as applied to the response rate of one group who received the challenger offer. In this case, there are actually two response rates, one for the champion and one for the challenger. Are these response rates different? Notice that the observed rates could be different (say 5.0 percent and 5.001 percent), but these could be indistinguishable from each other. One way to answer the question is to look at the confidence interval for each response rate and see whether they overlap. If the intervals do not overlap, then the response rates are different.

This example investigates a range of response rates from 4.5 percent to 5.5 percent for the champion model. In practice, a single response rate would be known. However, investigating a range makes it possible to understand what happens as the rate varies from much lower (4.5 percent) to the same (5.0 percent) to much larger (5.5 percent).

The 95 percent confidence is 1.96 standard deviation from the mean, so the lower value is the mean minus this number of standard deviations and the upper is the mean plus this value. Table 5.2 shows the lower and upper bounds for a range of response rates for the champion model going from 4.5 percent to 5.5 percent.

5% С 4%

3% i= 2% ja 1% £ 0%

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Observed Response Rate

Figure 5.8 Statistics has proven that actual response rate on a population is very close to a normal distribution whose mean is the measured response on a sample and whose standard deviation is the standard error of proportion (SEP).




Table 5.2 The 95 Percent Confidence Interval Bounds for the Champion Group

1 RESPONSE

SIZE

95% CONF

95% CONF * SEP

LOWER

UPPER 1

4.5%

900,000

0.0219%

1.96

0.0219%*1.96=0.0429%

4.46%

4.54%

4.6%

900,000

0.0221%

1.96

0.0221 %*1.96=0.0433%

4.56%

4.64%

4.7%

900,000

0.0223%

1.96

0.0223%* 1.96=0.0437%

4.66%

4.74%

4.8%

900,000

0.0225%

1.96

0.0225%* 1.96=0.0441 %

4.76%

4.84%

4.9%

900,000

0.0228%

1.96

0.0228%* 1.96=0.0447%

4.86%

4.94%

5.0%

900,000

0.0230%

1.96

0.0230%* 1.96=0.0451 %

4.95%

5.05%

5.1%

900,000

0.0232%

1.96

0.0232%* 1.96=0.0455%

5.05%

5.15%

5.2%

900,000

0.0234%

1.96

0.0234%* 1.96=0.0459%

5.15%

5.25%

5.3%

900,000

0.0236%

1.96

0.0236%* 1.96=0.0463%

5.25%

5.35%

5.4%

900,000

0.0238%

1.96

0.0238%* 1.96=0.0466%

5.35%

5.45%

5.5%

900,000

0.0240%

1.96

0.0240%*1.96=0.0470%

5.45%

5.55%

Response rates vary from 4.5% to 5.5%. The bounds for the 95% confidence level are calculated usingl .96 standard deviations from the mean.



Based on these possible response rates, it is possible to tell if the confidence bounds overlap. The 95 percent confidence bounds for the challenger model were from about 4.86 percent to 5.14 percent. These bounds overlap the confidence bounds for the champion model when its response rates are 4.9 percent, 5.0 percent, or 5.1 percent. For instance, the confidence interval for a response rate of 4.9 percent goes from 4.86 percent to 4.94 percent; this does overlap 4.86 percent-5.14 percent. Using the overlapping bounds method, we would consider these statistically the same.

Comparing Results Using Difference of Proportions

Overlapping bounds is easy but its results are a bit pessimistic. That is, even though the confidence intervals overlap, we might still be quite confident that the difference is not due to chance with some given level of confidence. Another approach is to look at the difference between response rates, rather than the rates themselves. Just as there is a formula for the standard error of a proportion, there is a formula for the standard error of a difference of proportions (SEDP):

This formula is a lot like the formula for the standard error of a proportion, except the part in the square root is repeated for each group. Table 5.3 shows this applied to the champion challenger problem with response rates varying between 4.5 percent and 5.5 percent for the champion group.

By the difference of proportions, three response rates on the champion have a confidence under 95 percent (that is, the p-value exceeds 5 percent). If the challenger response rate is 5.0 percent and the champion is 5.1 percent, then the difference in response rates might be due to chance. However, if the champion has a response rate of 5.2 percent, then the likelihood of the difference being due to chance falls to under 1 percent.

affected the result. There may be many other factors that we need to take into consideration to determine if two offers are significantly different. Each group must be selected entirely randomly from the whole population for the difference of proportions method to work.

SEDP =



Confidence intervals only measure the likelihood that sampling



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