(a) Right-tailed $Z$-test: $Z=3.54$ exceeds the $1\%$ critical value $2.33$ → reject $H_0$, claim supported. (b) Quarter averages $74.4,71.6,72.4,72.8$; grand average $72.8$; indices $=\frac{\text{Q avg}}{72.8}\times100$.

Part (a) — Z-test for a single mean

Given $\mu=1800,\ \sigma=100,\ n=50,\ \bar x=1850$, level $\alpha=0.01$.

Step 1 — Hypotheses (claim: strength increased):

$H_0:\mu=1800\quad\text{vs}\quad H_1:\mu>1800\ \text{(right-tailed)}.$

Step 2 — Test statistic:

$Z=\frac{\bar x-\mu}{\sigma/\sqrt{n}}=\frac{1850-1800}{100/\sqrt{50}}=\frac{50}{100/7.071}=\frac{50}{14.142}=3.54.$

Step 3 — Decision. Critical value at $1\%$ (one-tailed) $=2.33$. Since calculated $|Z|=3.54>2.33$, we reject $H_0$.

The sample gives strong evidence that the mean breaking strength has increased.

Part (b) — Seasonal indices by the simple-average method

Step 1 — Quarter totals and averages (5 years):

Step 2 — Grand (overall) average of the quarterly averages:

$\bar{\bar{y}}=\frac{74.4+71.6+72.4+72.8}{4}=\frac{291.2}{4}=72.8.$

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