Question

In a hospital’s shipment of 3,500 insulin syringes, 14 were unusual due to defects. (a) At α = .05, is this sufficient evidence to reject future shipments from this supplier if the hospital’s quality standard requires 99.7 percent of syringes to be acceptable? State the hypotheses and decision rule. (b) May normality of the sample proportion p be assumed? (c) Explain the effects of Type I error and Type II error. (d) Find the p-value. For full credit, the statement of effects for Type I and Type II errors must be at least a full sentence.

Please show all of your work and explain.

Answer 1

Proportion of acceptable units (p₁) = 1 - 14/3500 = 1 - 0.004 = 0.996

n = 3500

a)

alpha = 0.05

Z_Critical = +/- 1.96

Null and Alternate Hypothesis

H₀: p = 0.997

H_a: p <> 0.997

Test Statistic

z = (p₁ – p₀) / {p₀*(1- p₀)/n}^1/2 = (0.996-0.997)/ (0.997*(1-0.997)/3500)^1/2 = -1.08

p-value = TDIST(1.08,3500-1,2) = 0.279

Result

Since the p-value is greater than 0.05, we fail to reject the null hypothesis.

Conclusion

There is no sufficient evidence to reject that claim that the proportion of suppliers acceptable units is 99.7%

b)

Since the sample size is large enough (greater than 30) , Yes, we can assume normality assumptions to hold true

c)

a type I error is the rejection of a true null hypothesis while a type II error is the non-rejection of a false null hypothesis.

d)

p-value = 0.279

Type I Error: If in truth the proportion of acceptable units had been 0.997 and if we had incorrectly reject the claim, then we would have made a Type 1 Error

Type II Error: If in truth the proportion of acceptable units had not been 0.997 and if we had incorrectly failed to reject the claim, then we would have made a Type II Error