Question

A watch manufacturer believes that 60% of men over age 50 wear watches. So, the manufacturer took a simple random sample of 275 men over age 50 and 170 of those men wore watches. Test the watch manufacturer's claim at α = .05.

Answer 1

Solution

Let X = number of men over age 50 who wear watches in asample of 275 men.

Then, X ~ B(n, p), where n = sample size and p = the proportion of men over age 50 who wear watches    in the population.

Claim :

60% of men over age 50 wear watches

Hypotheses:

Null H₀ : p = p₀   = 0.6 [i.e., 60%] Vs Alternative H_A : p ≠ p₀

Test Statistic:

Z = (p_hat - p₀)/√{p₀(1 - p₀)/n}

Where

p_hat = sample proportion and

n = sample size.

Calculations:

p0	0.60
n	275
x	170
phat	0.6182
Zcal	0.6155
α	0.05
Zcrit	1.9600
p-value	0.5383
p0(1-p0)/n	0.0009
sqrt	0.0295
1 - (α/2)	0.9750
1 - α	0.9500
1-pvalue	0.7309

Distribution, Significance Level, α Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper (α/2)% of N(0, 1), and

p-value = P(Z > | Zcal |)

Using Excel Function: Statistical NORMSINV and NORMSDIST these are found as shown in the above table.

Decision:

Since | Zcal | < Zcrit, or equivalently, since p-value > α, H₀ is accepted.

Conclusion :

There is enough evidence to suggest that the claim is valid.

Hence we conclude that the manufacturer’s claim is valid. Answer

DONE