Can Both Hypotheses be Accepted with a Confidence Level of 68%?

[cutesteph]

1. Homework Statement
Suppose you receive calls that follow a Poisson process model Y(t).
There are two hypotheses, Hypothesis1: E[Y(t)] = λ1t = 70t and Hypothesis 2: E[Y(t)] = λ2t = 75t. Let t = 30 the number of calls be 2175.

Find and compute a significance level α such that both Hypothesis1 and Hypothesis2 are accepted.
2. Homework Equations

α = P( (u - E[Y] /stdY > c / stdY) 3. The Attempt at a Solution

 

[Simon Bridge]

Good question - how are you attempting the problem?
Presumably you've done problems involving confidence intervals and/or poisson distributon before?

 

[cutesteph]

Opps, I forgot to type my approach. Since n is big, we can approximate the poisson with the normal.
The variance of a poisson is the same as the expected value.
Or am I suppose to use a two sided test?

 

[Simon Bridge]

Well to 68% confidence limits, would you accept both hypotheses?

 

[blue_raver22]

I would first point out that accepting both hypotheses with a confidence level of 68% is not a common practice in scientific research. Typically, a higher confidence level (such as 95% or 99%) is required to accept a hypothesis.

That being said, let's consider the given information and attempt to find a significance level α that would result in accepting both hypotheses.

The first step would be to calculate the expected number of calls for each hypothesis at t = 30. For Hypothesis 1, we have E[Y(30)] = λ1(30) = 70(30) = 2100. For Hypothesis 2, we have E[Y(30)] = λ2(30) = 75(30) = 2250.

Next, we need to calculate the standard deviation of Y at t = 30. The Poisson distribution has a standard deviation equal to the square root of the mean, so stdY = √(E[Y(30)]) = √(2175) ≈ 46.62.

Now, we can use the given formula for α to solve for the critical value c. Plugging in the values we have calculated, we get:

α = P((2175 - 2175) / 46.62 > c / 46.62) = P(0 > c / 46.62)

Since the probability of a negative value is 0, we can set c = 0 and still satisfy the inequality. Therefore, the significance level α is equal to 0.

This means that with a significance level of 0, both hypotheses can be accepted. However, as mentioned before, a significance level of 0 is not a common practice in scientific research. In order to have a higher confidence in our results, we would need to choose a higher significance level and potentially reject one or both of the hypotheses.