Question

A study followed 20 infants to find out when they speak their first word of English (in an English-speaking household). They found that the children were 13 months old on average when they spoke their first word; the data had a standard deviation of 4.93 months. What is the average age at which the population of children speak their first English word?

First answer each of these questions:

i) Does this call for a confidence interval or a hypothesis test?

ii) Is this 1 sample or 2 samples?

iii) Is this about mean(s) or proportion(s)?

iv) If this is about mean(s), do you know the SD of the population (σ — or σ1 & σ2 )? (If this is about proportions, skip this question.)

b) What are the population parameter(s), and what are the sample statistic(s)? Provide symbols, and say what they represent in the context of the question. (For example, "µ is the mean completion time of the population, and x̄ is the mean completion time for the sample".) You may want to give the statistic values here.

c) Complete the question. Details depend on the kind of problem:

• For a confidence interval:

i) State the confidence level. (If it is not given, make a reasonable choice.)

ii) Give the formula for the margin of error (symbols only, no numbers!).

iii) Calculate the margin of error (show your work!).

iv) State the confidence interval in a complete sentence (in words!), in the context of the original problem. (You may use whichever form you prefer.)

• For a hypothesis test:

i) State the significance level (alpha). (If it is not given, make a reasonable choice.)

ii) Give the formula for the test statistic (z or t) (symbols only, no numbers!).

iii) State the null and alternative hypotheses. Use symbols, and state them in the context of the original problem. (A sketch is optional, but very useful.)

iv) Calculate the test statistic (z or t) (show your work!), and determine the p-value.

v) State your conclusion in a complete sentence (in words!), in the context of the original problem. Your conclusion should state whether or not you reject the null hypothesis, and what this says about the original question.

Answer 1

(a)

(i) This calls for a confidence interval

(ii) This is 1 sample t test

(iii) This is about mean

(iv) We do not know the SD of population

(b)

(i) $\mu$ = Mean age at which the population of children speak their first English word

(ii) $\bar{x}$ = Mean age at which the sample of children speak their first English word = 13

(c)

(i) Confidence level = $\alpha$ = 0.05 (not given)

(ii) Margin of Error =E = Z * SE

(iii) SE = s/$\sqrt{n}$

= 4.93/$\sqrt{20}$ = 1.1024

From Table, critical values of Z= $\pm$ 1.96

So,

E = 1.96 X 1.1024= 2.1607

(iv)

Confidence Interval:

13 $\pm$ 2.1607

= ( 10.8393, 15.1607)

There is 0.95 probability that mean age at which the population of children speak their first English word lies in the interval:

10.8393 < $\mu$ < 15.1607