Question

Smartphones: A Pew Research report indicated that 73% of teenagers aged 13–17 own smartphones. A random sample of 150 teenagers is drawn. a. Find the probability that more than 70% of the sampled teenagers own a smartphone. b. Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.76 and 0.80.

Answer 1

Since sample size is large we have to apply Normal approximation to the Binomial distribution.

Let us assume probability of usage of smartphone as probability of success.

Thus, probability of success: p = 0.73

Probability of failure: q = 1 - p = 1 - 0.73 = 0.27

Sample size: n = 150

Here n*p = 150*0.73 = 109.5 > 5

and n*q = 150*0.27 = 40.5 > 5

Since both np and nq are greater than 5 we can Normal approximation to the Binomial for this question.

Find mean of the distribution:

Mean of Binomial distribution is given by formula:

Find standard deviation of the distribution:

Standard deviation of Binomial distribution is given by formula:

a) We have to find a probability that more than 70% of the sampled teenagers own a smartphone.

70% of 150 = 0.70*150 = 105

Therefore we have to find probability that more than 105 teenagers own a smartphone.

Applying continuity correction factor rule we get,

Convert 'x' into 'z' score:

From z score table we get,

Therefore probability that more than 70% of the sampled teenagers own a smartphone is 0.7704 or 77.04%

b) We have to find a probability that proportion of smartphone owning teenagers is between 0.76 to 0.80

76% of 150 = 0.76*150 = 115

80% of 150 = 0.8 * 150 = 120

That means we have to find probability that number of smartphone users is between 115 and 120.

i.e.

Use continuity correction rule:

Converting 'x' into 'z' score we get,

From z table we get,

Therefore probability that proportion of the sampled teenagers who own a smartphone between 0.76 to 0.8 is 0.1028 ot 10.28%