Question

10. A radio manufacturer claims that 65% of teenagers (13-16 years old) have their own head phones. A researcher wishes to test the claim and selects a random sample of 80 teenagers. She finds that 57 have their own head phones. At α-0.05, should the claim be rejected? Construct a corresponding confidence interval for the population parameter.

Answer 1

H₀: P = 0.65

H₁: P $\neq$ 0.65

$\widehat p$ = 57/80 = 0.7125

The test statistic z = ( $\widehat p$ - P)/sqrt(P(1 - P)/n)

= (0.7125 - 0.65)/(0.65 * 0.35/80)

= 1.17

At alpha = 0.05, the critical values are z_0.025 = +/- 1.96

Since the test statistic value is not greater than the upper critical value (1.17 < 1.96), so we should not reject the null hypothesis.

So at alpha = 0.05, there is not sufficient evidence to reject the claim.

The 95% confidence interval is

$\widehat p$ +/- z_0.025 * sqrt( $\widehat p$ (1 - $\widehat p$ )/n)

= 0.7125 +/- 1.96 * sqrt(0.7125 * (1 - 0.7125)/80)

= 0.7125 +/- 0.0992

= 0.6133, 0.8117