Question

In a large university, 70% of the students live in the dormitories. A random sample of 110 students is selected for a particular study.The probability that the sample proportion of students living in the dormitories falls in between 0.6 and 0.8 equals a.0.9780 b.0.9318 c.0.9365 d.1.6450

Answer 1

Answer:

The correct option is (a) 0.9780.

Step-by-step explanation:

According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this sampling distribution of sample proportion is:

 $\mu_{\hat p}=p$

The standard deviation of this sampling distribution of sample proportion is:

 $\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$

As the sample selected is quite large, i.e. n = 110 > 30, the central limit theorem can be applied to approximate the sampling distribution of sample proportion by a Normal distribution.

The mean and standard deviation are:

$\mu_{\hat p}=p=0.70\\\\\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.70(1-0.70)}{110}}=0.044$

Compute the probability that the sample proportion of students living in the dormitories falls in between 0.60 and 0.80 as follows:

$P(0.60$

                              $=P(-2.27$

*Use a z-table.

Thus, the probability that the sample proportion of students living in the dormitories falls in between 0.60 and 0.80 is approximately equal to 0.9780.

The correct option is (a).