Report states that the mean cost of raising a child from birth to age 2 in a rural area is $\$8,000$ with a standard deviation of $\$1500.$ A random sample of 900 children shows that the mean cost is $\$7925.$ At $\alpha = 5\%,$ decide if there is enough evidence to conclude that the mean cost is not $\$8000.$
I have
- $H_0: µ = 8000$
- $H_a: µ ≠ 8000$
$Z = (7925 - 8000) / 1500 = -0.05$
$P(Z \le -0.05) + P(Z \ge 0.05) = 2P(Z \ge 0.05) = 2[0.5-0.0199] = 0.9692.$
Rejection region is both $$Z_{statistic} \lt -Z_{critical}$$ and $$Z_{statistic} \gt Z_{critical}$$ Find $Z_{critical}$ using a two sided level of significance $\frac{\alpha}{2}$.
$Z_{critical} = 1.96$
In this case you $-0.05 \gt -1.96$ and $\lt 1.96$ and "hence you do not reject the claim" and conclude that there is not sufficient evidence that the mean cost is not 8000.