Question

Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of 36 children who were identified as gifted children soon after they reached the age of four years old. One recorded characteristic was the age in months that each child was first able to count to ten. The sample mean was 30.69 months with a standard deviation of 4.31 months. Suppose you read online that children first count to ten successfully when they are 32 months old, on average. You decide test the hypothesis that the average age at which gifted children first count to ten is different from 32 months. What would the test statistic (i.e., Z-score) be for the null hypothesis in this case?

Answer 1

Z-score = -1.8914

Explanation:

We are given the following in the question:

Population mean, μ = 32 months

Sample mean,
`\bar{x}` = 30.69 months

Sample size, n = 36

Alpha, α = 0.05

Population standard deviation, σ = 4.31 months

First, we design the null and the alternate hypothesis

`H_(0): \mu = 32\text{ months}\\H_A: \mu \\eq 32\text{ months}`

The null hypothesis states that the average age at which gifted children first count to ten is same as population average 32 months and the alternate hypothesis states that average age at which gifted children first count to ten is different from 32 months.

We use Two-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(30.69 - 32)/((4.32)/(√(36)) ) = -1.8914`