**Data and T-Test correlation**

**1. Above is the monthly birthrate (per 1,000 US populations) from 1940-1947. A total of n=96 months are represented. Let ρ be the correlation between birth rate and time.
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** The sample correlation is r=0.6191. Test at the 0.05 significance level the null ****hypothesis H0: ρ=o versus HA: ρ≠o. Report the test statistic and P-value.**

We shall conduct a t-test:

t=(r√(n-2))/√(1-r^2 )= 7.643326

The t-statistic has degrees of freedom n-2.

p-value=8.89×〖10〗^(-12)

Hence, we reject H0 in favor of alternative, and hence, we conclude that correlation is statistically significant.

** Is there an overall linear trend in the data? What does the positive correlation coefficient indicate in terms of the trend in birth rates over time?
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There is an overall trend in the data, but it is not linear. There are two shapes and the part-wise linear line can be fitted.

** Describe two non-linear trends in the data and what might be the demographic ****historical causes of those nonlinear trends.**

The trend we see is increasing from 40-43 then decreasing till1946. Then a sharp increase. This may be due to the world war going around at the time**.**

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** The plot above shows the brain weight (grams) versus body weight (kilograms) of n=62 species of mammals. Both variables have been transformed to the log scale. Define the regression model **

**y=α+β*𝓍+e, **

**'where y is the log(BrainWt), x is log(BodyWt), and e is random error.**

**a. According to the regression model from the computer output, what is the average change in the log(BrainWt) associated with an increase of one unit in the log(BodyWt)?
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The regression model is:

(log(BrainWt)) ̂=2.13479+0.751686×log(BodyWt)

**b. According to the regression model, what is the average brain weight for a 59kilogram mammal?**

log(brain wt)= 2.13479+0.751686×log〖(59)=5.2〗

Hence, brain wt=181.27gms

**c. Test the null hypothesis that β=0 versus the alternative β≠0 at the 0.05significance level. Give the test statistic, degrees of freedom, p-value, and your conclusion.**

We can carry out t-test

t=β ̂/se(β) =0.751686/0.0284635= 26.40877

The test statistic has a t-distribution with 60 degrees of freedom.

P-value is < 0.0001

Conclusion: The body weight can significantly predict brain weight and the strength of the linear relationship between them is statistically significant.

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