Question

A large survey of people throughout the world has indicated that people claim that their favorite color is red. In fact, about 20% of the population claim that red is their favorite color. Suppose a random sample of n = 35 college students was surveyed and r = 8 of them said that red is their favorite color. Does this indicate that the color preference of all college students is different (either way) from that of the general population? Use level of significance = 0.02 . Run hypothesis-testing.

Answer 1

Solution: For the given question we construct our null and alternative hypotheses as:

H0: p = 0.20 vs Ha: p $\neq$ 0.20 [Since we are to see if  the color preference of all college students is different (either way) from that of the general population, so a two-taied test ]

[Here p= unknown population proportion of the college students whose favourite color is red]

The test statistic for this is Z = (p_hat - p0)/sqrt(p0*(1-p0)/n) ~ N(0,1) under H0.

where p0 = hypothetical value of the unknown population proportion, n= sample size, p_hat = sample proportion.

Here p0 = 0.2, n = 35, p = 8/35

We reject H0 if |Z(observed)| > tau(alpha/2) where tau(alpha/2) is the upper alpha/2 point of a standard normal distribution.

Here Z(observed) = 0.4225771 and tau(alpha/2) = 2.326348

So, we see that |Z(observed)| < tau(alpha/2), hence we fail to reject H0 and conclude at a 2% level of significance on the basis of the given sample measures that there is not enough evidence to say that the color preference of all college students is different (either way) from that of the general population.