Question

Use the following information for the next three problems.  Suppose that  = 30% of the students at a large university must take a statistics course.  Suppose that a random sample of 50 students is selected.  Let  = the percent of students in the sample who must take statistics.

Different samples will produce different values of .  In order for the values of  to vary according to a normal model, we need to check two conditions.  Which two of the following need to be checked?

	a.	and
	b.	The sample size  is large ().
	c.	The sample observations are independent of each other.
	d.	The population distribution is normal in shape.

2. 95% of all samples will produce a  between __________ and __________.

	a.	0.170 and 0.430
	b.	0.105 and 0.495
	c.	0.280 and 0.320
	d.	0.235 and 0.365

3. What is the chance that more than 38% of the students in a sample must take statistics (i.e. what is the chance that  > 0.38)?

	a.	1.23
	b.	0.8907
	c.	0.6480
	d.	0.1093

Answer 1

The sample size is large ().
The sample observations are independent of each other.

(2)
sample proportion, = 0.3
sample size, n = 50
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.3 * (1 - 0.3)/50) = 0.065

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.3 - 1.96 * 0.065 , 0.3 + 1.96 * 0.065)
CI = (0.170 , 0.430)

Option A

3)
Here, μ = 0.3, σ = 0.065 and x = 0.38. We need to compute P(X >= 0.38). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.38 - 0.3)/0.065 = 1.23

Therefore,
P(X >= 0.38) = P(z <= (0.38 - 0.3)/0.065)
= P(z >= 1.23)
= 1 - 0.8907 = 0.1093

Option D