Question

To properly treat patients, drugs prescribed by physicians must not only have a mean potency value as specified on the drug's container, but also the variation in potency values must be small. Otherwise, pharmacists would be distributing drug prescriptions that could be harmfully potent or have a low potency and be ineffective. A drug manufacturer claims that his drug has a potency of 5 ± 0.1 milligram per cubic centimeter (mg/cc). A random sample of four containers gave potency readings equal to 4.94, 5.09, 5.02, and 4.89 mg/cc.

(a) Do the data present sufficient evidence to indicate that the mean potency differs from 5 mg/cc? (Use α = 0.05.)

State the test statistic (round to three decimal places)

(b) Do the data present sufficient evidence to indicate that the variation in potency differs from the error limits specified by the manufacturer? (HINT: It is sometimes difficult to determine exactly what is meant by limits on potency as specified by a manufacturer. Since he implies that the potency values will fall into the interval 5 ± 0.1 mg/cc with very high probability—the implication is almost always—let us assume that the range 0.2, or 4.9 to 5.1, represents 6σ, as suggested by the Empirical Rule. Use α = 0.05.)

- State the test statistic. (Round your answer to three decimal places.)

χ² =

- State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.)

χ² > ___

χ^{2 < ____}

Answer 1

Given

n = 4

Random samples 4.94,5.09,5.02 , 4.89

Mean = 4.985

Standard deviation = 0.076

a)

$\mu$ = 5

α = 0.05

t = ($\bar{x}-\mu$)/(sd/sqrt(n))

t = (4.985 - 5) / (0.076 / sqrt (4) )

t = -0.3947

t- critical = 3.18

t < t-critical

So, there is no sufficient evidence to indicate that the mean potency differs from 5 mg /cc

b)

6σ=0.2

σ = 0.2/3 = 0.0667

c)

chi-square = (n-1)s²/σ² = (4-1)0.076² / 0.0667² = 3.8949

d)

χ² > 9.348

χ^{2 < 0.216}