Problem:

The total fertility rate (TFR), sometimes also called the fertility rate, is the average number of children that would be born to a woman over her life time.In 2000, the TFR was 2.06 in the United States. Some fifteen years later, in a2015 study of 225 women, researchers calculated that in their sample, the TFR was 1.78 with a standard deviation of 2. The researchers want to know whetherTFR decreased from 2005 to 2015.

Requirement:

1. Write down and explain the null and alternative hypothesis.

2. Write down the test statistics and explain in detail how it is distributed when the null hypothesis is true. Also explain in detail how the test statistic behave when the null hypothesis is false.

3. What is Type I Error? State the significance level you will use in designing the test. And show the rejection intervals.

4.  Show in a graph the test rejection you would use. Be sure to label the distributionof the test statistic when the null hypothesis is true and indicatewhere the probability of Type I Error is depicted in your graph.

AppendixPercentiles for standard normal distribution

1. Quantile for N(0, 1) with 0.1 probability = -1.2815522.

Quantile for N(0, 1) with 0.9 probability = 1.2815523.

Quantile for N(0, 1) with 0.05 probability = -1.6448544.

Quantile for N(0, 1) with 0.95 probability = 1.6448545.

Quantile for N(0, 1) with 0.025 probability = -1.9599646.

Quantile for N(0, 1) with 0.975 probability = 1.959964

Percentiles for t distribution1.

Quantile for t225!1 with 0.1 probability = -1.2853422.

Quantile for t225!1 with 0.9 probability = 1.2853423.

Quantile for t225!1 with 0.05 probability = -1.6516854.

Quantile for t225!1 with 0.95 probability = 1.6516855.

Quantile for t225!1 with 0.025 probability = -1.9706116.

Quantile for t225!1 with 0.975 probability = 1.9706117.

Quantile for t100!1 with 0.1 probability = -1.2901618.

Quantile for t100!1 with 0.9 probability = 1.2901619.

Quantile for t100!1 with 0.05 probability = -1.66039110.

Quantile for t100!1 with 0.95 probability = 1.66039111.

Quantile for t100!1 with 0.025 probability = -1.98421712.

Quantile for t100!1 with 0.975 probability = 1.984217