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Samples are taken from a specific batch of drug and randomly divided into two groups of tablets. One group is assayed by the manufacturer's own quality control laboratories. The second group of tablets is sent to a contract laboratory for identical analysis.

Manufacturer results: 101.1, 100.6, 98.8, 99.0, 100.8, 98.7
Contract Lab results: 97.5, 101.1, 99.1, 98.7, 97.8, 99.5.

Is there a significant difference between the results generated by the two labs? Use a t-test to compare the two groups.

Answer :

To determine whether there is a significant difference between the results generated by the manufacturer's laboratory and the contract laboratory, we can use a t-test. A t-test compares the means of two samples to see if they are statistically different from each other.

Step-by-Step Guide:

  1. State the Hypotheses:

    • Null Hypothesis (H₀): There is no significant difference between the two lab results, i.e., the means are equal.
    • Alternative Hypothesis (H₁): There is a significant difference between the two lab results, i.e., the means are not equal.

  2. Collect Data:

    • Manufacturer results: 101.1, 100.6, 98.8, 99.0, 100.8, 98.7
    • Contract Lab results: 97.5, 101.1, 99.1, 98.7, 97.8, 99.5

  3. Calculate the Means and Standard Deviations:

    • Mean (Manufacturer) = (101.1 + 100.6 + 98.8 + 99.0 + 100.8 + 98.7) / 6

    • Mean (Contract Lab) = (97.5 + 101.1 + 99.1 + 98.7 + 97.8 + 99.5) / 6

    • Calculate standard deviation for each group.

  4. Perform the T-Test:
    Use the formula for the t-test for two independent samples:

    [tex]t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}[/tex]

    Where:

    • [tex]\bar{x}_1[/tex] and [tex]\bar{x}_2[/tex] are the means of the two samples.
    • [tex]s_1^2[/tex] and [tex]s_2^2[/tex] are the variances of the two samples.
    • [tex]n_1[/tex] and [tex]n_2[/tex] are the sample sizes.

  5. Determine the Degrees of Freedom and Critical t-value:

    • Combine the degrees of freedom: [tex]df = n_1 + n_2 - 2[/tex]
    • Use a t-table to find the critical t-value for the desired significance level (commonly 0.05) with the calculated degrees of freedom.

  6. Compare the Calculated t-value with the Critical t-value:

    • If the calculated t is greater than the critical t-value, reject the null hypothesis.
    • If the calculated t is less than or equal to the critical t-value, do not reject the null hypothesis.

Conclusion:

After performing the calculations, you can decide if the difference between the two laboratories' results is statistically significant. If the test indicates significance, this suggests the laboratories might produce different results under similar conditions, possibly due to different testing procedures or equipment calibration. Without running the exact calculations, the final answer cannot be definitively provided, but the outlined steps will guide you to the solution.

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