Question

As a student project Natalie would like to see whether there is statistical evidence Earth is warmer today versus 30 years ago. She randomly samples 100 locations around the world and collects the annual average high for each location for the years 1985 and 2015. She proceeds to compare the differences among the time periods with respect to location. What procedure should be used to test whether the average difference in temperature between the locations is greater than zero

Answer 1

Natalie would use a "test of difference between two means".

Explanation:

First step: State the hypotheses of the test as follows;

`H_(0) : U_(1) -U_(2) = 0\\H_(1) : U_(1) -U_(2) \geq 0`

Where
`U_1 ,U_2` are the Population means of years 1985 and 2015 respectively.

Second step: Determine the level of significance to be used for the testing. A 5% level of significance or 10% level of significance could be used.

Third step: Calculate the sample means and Variances for the two populations as follows;

`m_1 = (sum ofx_(1985)i)/(n) \\m_2=(sumofx_(2015)i)/(n)`

where
`m_1,m_2` are the annual average high warmth for the years 1985 and 2015 respectively.

`s_1= (sum(x_i-m_1)^(2) )/(n-1) \\s_2=(sum(x_i-m_2)^(2) )/(n-1)`

where
`s_1,s_2` are the respective sample variances for 1985 and 2015

and
`n=100`

Step three: The test statistic is computed as follows;

`Z=\frac{m_1-m_2}{\sqrt{(s_1)/(n)+(s_2)/(n) } }`

Step four: The Critical point for acceptance is calculated using the normal table and chosen level of significance. It is represented as
`Z_\alpha`

Step five: The decision rule is specified as follows:

Reject
`H_0` if
`Z\geq Z_ \alpha`, otherwise do not reject.

Step six: Conclusion is made based on the outcome of the test.

Note:
`\geq` used here represents "greater than"

Answer 2

She should use the variance and standard deviation

Explanation:

In order to make her results statistically significant, the student has to use statistical methods to check the results. In this way, the student will be able to determine whether there is a clear correlation between her results. In addition, using the standard deviation would be a useful tool to check whether her data will not be very much scattered throughout the distribution. By using the standard deviation, she will be able to see how far her data moves away from the mean or average.