At what age do infants speak their first word of English? Here are data on 20 children (ages in months):

15, 26, 10, 9, 15, 20, 18, 11, 8, 20, 7, 9, 10, 11, 11, 10, 12, 17, 11, 10

We are told by the American Academy of Pediatrics that the standard deviation of the mean age in the population of all normal children in America is [tex]\sigma = 4.5[/tex].

1. Is there good evidence that the mean age at first word among all normal children is greater than 12 months (one year)? State the null and alternative hypotheses. Carry out a test to answer the question.

2. Based on the first question, find an interval with 95% confidence level for the mean age at first word among all normal children. Does your interval explain your answer for question 1?

The mean age at which infants speak their first word in English is determined through a hypothesis test using the given data. The hypothesis test concludes that there is not enough evidence to suggest that the mean age is greater than 12 months.

Explanation:

The null hypothesis is that the mean age at which infants speak their first word is not greater than 12 months. The alternative hypothesis is that the mean age at which infants speak their first word is greater than 12 months.

To test this hypothesis, we can perform a one-sample t-test. We calculate the sample mean of the given data, which is 12.55 months, and the standard deviation of the sample, which is 4.86 months. Using these values, we can calculate the t-statistic, which is approximately 0.30. The p-value corresponding to this t-statistic is approximately 0.384, which is greater than the significance level of 0.05.

Therefore, based on the test, we cannot reject the null hypothesis. There is not enough evidence to suggest that the mean age at which infants speak their first word is greater than 12 months.