Question

La Leche League International reports that the mean age of weaning a child from breastfeeding is age four to five worldwide. In America, most nursing mothers wean their children much earlier. Suppose a random survey is conducted of 21 U.S. mothers who recently weaned their children. The mean weaning age was 8 months (2/3 year) with a standard deviation of 3 months. Conduct a hypothesis test to determine if the mean weaning age in the U.S. is less than four years old. Conduct a hypothesis test at the 5% level.

Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

1-state the null and alternative hypothesis.

2-In words, state what your random variable X represents

3- State the distribution to use for the test. (Enter your answer in the form z or t_df where df is the degrees of freedom.)

4- What is the test statistic?

5- what is the p-value? and what does it mean?

6- Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.

(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
α =

(ii) Decision:

reject the null hypothesis or

do not reject the null hypothesis    

(iii) Reason for decision:

Since α < p-value, we do not reject the null hypothesis.

Since α > p-value, we do not reject the null hypothesis.

Since α > p-value, we reject the null hypothesis.

Since α < p-value, we reject the null hypothesis.

(iv) Conclusion:

There is sufficient evidence to conclude that mean age at which American mothers wean their children is less than 4. or

There is not sufficient evidence to conclude that mean age at which American mothers wean their children is less than 4.   

Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Enter your answers in years. Round your answers to four decimal places.)

Answer 1

Solution:

Given:

The mean weaning age was 8 months with a standard deviation of 3 months.

First we need to convert given data of months into year.

Therefore,

Sample mean = 8/12 = 0.67

Sample standard deviation = s = 3/12 = 0.25

The sample size is n = 21.

This corresponds to a left-tail test, for which a t-test for one mean, with unknown population standard deviation will be used and sample size <30.

Null and Alternative Hypotheses:

The number of degrees of freedom are df = n-1 = 21-1=20

Rejection Region:
Given significance level = α = 0.05 and df = 20
P-value = 0 (Using standard t table)

Decision:

P-value = 0< α = 0.05 .
It is then concluded that the Null Hypothesis is not rejected.

Conclusion: It is concluded that the Null Hypothesis is not rejected. Therefore, There is sufficient evidence to conclude that mean age at which American mothers wean their children is less than 4 at the 0.05 significance level.

To find confidence interval, we need to know point estimate.

Hence, 95% confidence interval is,

0.556<μ<0.784.

Done