Question

Baby talk: In a sample of 80 children, the mean age at which they first began to combine words was 16.81 months, with a standard deviation of 9.65 months. Construct a 95% interval for the mean age at which they first begin to combine words.

Answer 1

A 95% confidence interval for the mean age at which children in the sample first began to combine words is calculated to be between approximately 14.72 months and 18.90 months.

Step-by-step explanation:

To construct a 95% confidence interval for the mean age, we use the formula:

`\[ \text{Confidence Interval} = \text{Mean} \pm \left( \text{Critical Value} * \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \]`

Given that the mean age
`(\(\bar{X}\))` is 16.81 months, the standard deviation
`(\(S\))` is 9.65 months, and the sample size
`(\(n\))` is 80, we need the critical value for a 95% confidence interval.

For a 95% confidence interval and a two-tailed test, the critical z-value is approximately 1.96.

`\[ \text{Confidence Interval} = 16.81 \pm \left( 1.96 * (9.65)/(√(80)) \right) \]`

Now, calculate the margin of error:

`\[ \text{Margin of Error} = 1.96 * (9.65)/(√(80)) \]`

`\[ \text{Margin of Error} \approx 1.96 * (9.65)/(8.94) \]`

`\[ \text{Margin of Error} \approx 1.96 * 1.080 \]`

`\[ \text{Margin of Error} \approx 2.11 \]`

Finally, construct the confidence interval:

`\[ \text{Confidence Interval} = (16.81 - 2.11, 16.81 + 2.11) \]`

`\[ \text{Confidence Interval} \approx (14.72, 18.90) \]`

Therefore, we can be 95% confident that the true mean age at which children first began to combine words is between approximately 14.72 months and 18.90 months.