Two Population Means with Unknown Standard Deviations Quiz

PDF page 546: The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t-test. The degrees of freedom formula was developed by Aspin-Welch.

PDF page 546: The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. To account for the variation, we take the difference of the sample means, , and divide by the standard error to standardize the difference. The result is a t-score test statistic. Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, The standard error is calculated as follows:

PDF page 546: • s1 and s2, the sample standard deviations, are estimates of σ1 and σ2, respectively,

PDF page 546: The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. To account for the variation, we take the difference of the sample means, , and divide by the standard error to standardize the difference. The result is a t-score test statistic. Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, The standard error is calculated as follows:

PDF page 546: The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. To account for the variation, we take the difference of the sample means, , and divide by the standard error to standardize the difference. The result is a t-score test statistic. Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, The standard error is calculated as follows:

PDF page 546: • s1 and s2, the sample standard deviations, are estimates of σ1 and σ2, respectively,

PDF page 546: • μ1 and μ2 are the population means.

PDF page 546: The number of degrees of freedom (df) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t-distribution with df as follows: Degrees of freedom

PDF page 547: When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample 2 2 variances (s1) and (s2) are not pooled. (If the question comes up, do not pool the variances.) It is not necessary to compute this by hand. A calculator or computer easily computes it. EXAMPLE 10.1

PDF page 547: When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample 2 2 variances (s1) and (s2) are not pooled. (If the question comes up, do not pool the variances.) It is not necessary to compute this by hand. A calculator or computer easily computes it. EXAMPLE 10.1

PDF page 546: ◦ if the sample sizes are small, the distributions are important (should be normal), and

PDF page 546: ◦ if the sample sizes are large, the distributions are not important (need not be normal)

PDF page 548: When the sum of the sample sizes is larger than 30 (n1 + n2 > 30), you can use the normal distribution to approximate the Student’s t.