L5 - Confidence Intervals

Question 1

How do you conduct a Hypothesis Test?

Answer 1

1-Hypothesis: Decide on your hypothesis. You need a null hypothesis and an alternative hypothesis.

2-Significance: Decide on the level of significance unless there is a good reason not to, people generally use 0.05 as the significance level also known as the alpha value. The significance level is the probability that you will say that the null hypothesis is wrong when really it is correct (TYPE I error).

3-Sample: Take a sample from the population to provide the statistics you need

4-P-value : Calculate the p-value. This is almost always done by a computer package. Based on the determination of suitable test statistics i.e. Z, t, Chi-squared, or F-statistics

5-Decision: Use the p-value to decide whether to reject the null hypothesis. If the p-value is less than significance level you chose earlier, you will reject the null hypothesis. The sample has given you evidence that the null hypothesis is wrong.

Question 2

What is the P-value?

Answer 2

• Let X be a random variable which can take on any real value between limits a and b.
• Let x_c be a real number which lies somewhere in the range a to b.
• The p-value of x_c is the probability that a random draw of X is

‘at least as extreme as x_c’ under a given null hypothesis.

Depending on the context, this may be:

• P(X>x_c), P(Xc) or P(X>x_c or X < -x_c)

Essentially, we are asking how far out in the tail of the distribution is the number x_c –> looking at the shaded area of the tails

_- For the case P(X> x_c) we can evaluate the p-value as the area under the tail of the distribution to the right of x_c, The p-value gives the probability of that the random variable X will be as least as large as the value given under the given null.

• For the case P(Xc) we evaluate the p-value as the area under the left tail of the distribution.

Question 3

What are Critical Values?

Answer 3

• A critical value is a value corresponding to a predetermined p-value. For example a 5% critical value is a value of the test statistic which would yield a p-value of 0.05 ( also called signficance level –> α
• Critical values are often set at 10%, 5% or 1% levels:

10% 1.282

5% 1.645

1% 2.326

• If the test statistic exceeds the critical value then the test is said to reject the null hypothesis at that particular critical value.
• Critical values are chosen so as to fix the probability of making a Type I error i.e. we choose a critical value which determines aparticular probability of falsely rejecting the null.
• This is known as the size of the test. Note that the smaller we make the probability of a Type I error then the larger is the probability of a Type II error.

Question 4

Why do we fix the size of a hypothesis test?

Answer 4

• The size of the test fixes the acceptable probability of making a type I error i.e.

P(Reject H₀|H₀ is true)

• Therefore if we fix the size of the test at 5% and repeat the test 100 time we would expect to make a Type I error in five cases. A test is said to be stricter the harder we make to reject the null hypothesis.
• For example, adopting a 1% significance level results in a stricter test. Similarly, adopting a two-sided alternative implies a stricter test for the same significance level.

Question 5

What is the Power of the test?

Answer 5

The power of a test is defined as:

Power = 1 - P(Type II error)

• Therefore the stricter is the test (i.e. the smaller is its size) then the lower is its power
• It is hard to attach a specific number to the power of a test but we can often rank different tests in terms of their relative power.
• If a number of different tests are available then we would normally choose the most powerful test.
• If we have a test against a specific alternative then we can put a numerical value on the power of the test.

Question 6

What is a confidence interval?

Answer 6

• The confidence interval is an interval estimate of a parameter of interest. It indicates an interval within which the unknown parameter is likely to lie.
• Confidence intervals are particularly associated with classical or frequentist statistical theory.
• A confidence interval is normally expressed in percentage terms e.g. ‘we are 95% confident that the population mean lies between the limits L and U’.

The way to interpret a 95% confidence interval is that if we were to repeat a random experiment 100 times then 95 of the interval estimates we calculate would contain the true population parameter.

• this calculation assumes that the estimate is normally distributed. This is a reasonable assumption in a large sample but may not be appropriate in small samples

Question 7

How can you derive a confidence interval for slope coefficent?

Answer 7

β(hat)~N(β, (σ²_u)/(Σ(X_i-X(bar))²)

(β(hat)-β)/ (σ_u)/sqrt(Σ(X_i-X(bar))²)~N(0,1)

This allows us to derive a condifence interval for a slope at 95% confidence:

p(-1.96 < (β(hat)-β)/ (σ_u)/sqrt(Σ(X_i-X(bar))²) <1.96) = 0.95

p(β(hat)-1.96 *(σ_u)/sqrt(Σ(X_i-X(bar))²) < β <β(hat)+1.96 *(σ_u)/sqrt(Σ(X_i-X(bar))²) = 0.95

Question 8

How do we perform a hypothesis test without the value of variance?

Answer 8

• using a t-test
• one-sample t-test is used to compare the mean of a population to a specified theoretical mean (μ).

(β(hat)-β)/ (σ_u)/sqrt(Σ(X_i-X(bar))²)

• we replace σ_u with an unbiased estimate then we can show that: (β(hat)-β)/ (σ_u(hat))/sqrt(Σ(X_i-X(bar))²)~ t_n-2
• Where (σ_u(hat))² = (1/N-2) Σ^N_i=1(Y_i -α(hat) -β(hat)X_i)²
• in a regression model degrees of freedom = n - p - 1 where n is number of events and p is the number of explanatory parameter in the model ( in this case it will probably be N - 2)
• Thus we have determined the small sample distribution of the OLS estimator for the slope coefficient. We can use this to test hypotheses about the unknown population parameter.

Question 9

What is the standard error of the slope coefficient?

Answer 9

SE(β(hat)= σ_u(hat)sqrt(1/Σ^N_i=1(X_i-X(bar))²)

Question 10

What is the standard error of the intercept?

Answer 10

SE(α(hat)= σ_u(hat)sqrt((1/N) + (X(bar))²/(Σ^N_i=1(X_i-X(bar))²

Question 11

How can we use the standard error test statistic?

Answer 11

H₀ : β = β(bar) –> β(hat)- β(bar)/SE(β(hat) ~ t_N-2

H₀ : α = α(bar) –> α(hat)- α(bar)/SE(α(hat) ~ t_N-2​