Question

An insurance company states that at least 86% of its claims are settled within 30 days. A consumer group selected a random sample of 73 of the company's claims to test this statement. They found that 60 of the claims were settled within 30 days. Does the consumer group have evidence to disbelieve the insurance company's claim?

1. Write the hypotheses to test if the rate of claims settled within 30 days is significantly less than 86%.

H0H0: The proportion of claims settled within 30 days  ? is equal to is less than is more than  86%.

HAHA: The proportion of claims settled within 30 days  ? is equal to is less than is more than  86%.

2. To setup a simulation for this situation, we let each claim be represented with a card. We take 100 cards,  black cards represent claims that are settled within 30 days and  red cards represent claims that take longer than 30 days to settle. Shuffle the cards and draw  ? with without  replacement  cards representing the random sample of claims. Calculate the proportion of  ? black red  cards in the  ? sample deck  and call it p^simp^sim . Repeat 10,000 times and plot the resulting sample proportions. The p-value will be the proportion of simulations where p^simp^sim is  ? less than or equal to less than or equal to unequal to   .

3. Use the One Proportion Resampling test app embedded here to perform the simulation. Draw at least 3000 samples. Report your p-value from the app here, rounded to four decimal places:

Use this external link to the One Proportion Resampling Test if the app does not load properly on your computer. (Right click to open in a new tab or window.)

4. What is the correct conclusion, based on your p-value?
A. The p-value is substantially larger than 0.05 and we should reject the null hypothesis.
B. The p-value is substantially smaller than 0.05 and we should not reject the null hypothesis.
C. The p-value is substantially larger than 0.05 and we should not reject the null hypothesis.
D. The p-value is substantially smaller than 0.05 and we should reject the null hypothesis.

Answer 1

1. H0: The proportion of claims settled within 30 days is equal to 86%.

HA: The proportion of claims settled within 30 days is less than 86%.

2. To setup a simulation for this situation, we let each claim be represented with a card. We take 100 cards, 86 black cards represent claims that are settled within 30 days and 14 red cards represent claims that take longer than 30 days to settle. Shuffle the cards and draw with replacement 73 cards representing the random sample of claims. Calculate the proportion of black cards in the sample and call it p^simp^sim . Repeat 10,000 times and plot the resulting sample proportions. The p-value will be the proportion of simulations where p^simp^sim is less than or equal to 0.8219.

3.

Observed	Hypothesized
0.8219	0.86	p (as decimal)
60/73	63/73	p (as fraction)
60.	62.78	X
73	73	n
	0.0406	std. error
	-0.94	z
	.1742	p-value (one-tailed, lower)

P-value = 0.1742.

4. C. The p-value is substantially larger than 0.05 and we should not reject the null hypothesis.