# Statistics tasks

An analytician is studying the profits of a company over the course of ten years. Let X be the number of years after the inital year the analytician is studying. Y is the profits in year X and the analytician believes the profits are a normal distribution stochastic(random) variable with expectation 𝐸(𝑌)=𝛽0+𝛽1𝑥 and standard deviation 𝜎=0.08. Based on these observations the analytician finds that:

𝑥¯=3.6 and 𝑦¯=4.26

𝑀=∑^9_𝑖=0 (𝑥𝑖−𝑥¯)^2=7.44,

∑^9_𝑖=0 (𝑥𝑖−𝑥¯)𝑦𝑖=4.96

(I added this as a picture too as I have a hard time writing formulas on here)

Questions:

1. **Estimate 𝛽0 og 𝛽1.**

2. Can it be proven that the yearly increase in profits, 𝛽1, is bigger than 0.5? Formulate fitting hypothesis and do a hypothesis test at 0,5% significance level. **Critical value= ****H1 or not H1?**

3.Find a 99% confidence interval for expected profits for x=5 **Upper confidence limit: ****Lower confidence limit:**

4. Find a 95% prediction interval for expected profits for x=1 **Upper prediction limit: ****Lower prediction limit:**

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