Question

5. The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 31% of students reported that their mothers had graduated from college. In 2000, responses from 8368 students found that this figure had grown to 32%. Is this evidence to suggest that the percentage of mothers who graduate from college is increasing? Conduct a hypothesis test to see if there is a significant difference at the 1% level.​

Answer 1

Based on the data, we cannot conclude that there is a significant difference in the percentage of mothers who graduate from college between 1996 and 2000.

Sure, let's conduct a hypothesis test to see if there is a significant difference in the percentage of mothers who graduate from college between 1996 and 2000.

**Null hypothesis:** p_1996 == p_2000 (There is no difference in the percentage of mothers who graduate from college between 1996 and 2000.)

**Alternative hypothesis:** p_1996 < p_2000 (The percentage of mothers who graduate from college is increasing.)

We will use a significance level of α = 0.01.

**Data:**

| Year | Sample size (n) | Proportion of mothers with college degree (p) |

|---|---|---|

| 1996 | 8368 | 0.31 |

| 2000 | 8368 | 0.32 |

**Pooled proportion:**

p_pooled = (n_1996 * p_1996 + n_2000 * p_2000) / (n_1996 + n_2000) = 0.315

**Test statistic:**

z = (p_2000 - p_pooled) / np.sqrt(p_pooled * (1 - p_pooled) * (1 / n_1996 + 1 / n_2000)) = 0.45

**Critical value:**

z_critical = stats.norm.ppf(1 - α / 2) = 2.326

**Decision:**

Since |z| < z_critical (0.45 < 2.326), we fail to reject the null hypothesis. There is not enough evidence to suggest that the percentage of mothers who graduate from college is increasing between 1996 and 2000.

**Conclusion:**

Based on the data, we cannot conclude that there is a significant difference in the percentage of mothers who graduate from college between 1996 and 2000.