One sided t test

CFA Level II
#1

Can someone please explain the one sided t test.

I always end up making mistakes with this.

Thanks.

#2

In a one-tailed t-test you can think about it as looking at one end of the distribution.

Where do you have trouble? Is it picking the appropriate critical value given the measure of reliability (type 1 error rate or confidence level)?

If, for example, you want to have the probability of a type one error (alpha) =0.05 (5%) for a one tailed test, you must notice that the entire probability is in one tail-- therefore, your critical value will be the same as if you were doing a 2-tailed test at an alpha of 0.10 (90% Confidence) doing the 2-tailed test at 0.10 means 0.05 is the probability in each tail. Let me know if this helped!

Take a look at this (or another) t-table and see if this helps now http://faculty.vassar.edu/lowry/PDF/t_tables.pdf

#3

Before looking at t-table, always ask a question. is this one tailed or two tailed?

If you null hypothesis includes ‘<=’ or ‘>=’ then its one tailed test.

If it contains only ‘=’ sign, its two tailed test.

I’ve missed this many times. I hope I remember in exam.

#4

No I have trouble finding out whether to reject the null or fail to reject the null.

#5

If your calculated test statistic is greater in magnitude than the critical value from the table (or if the p-value is smaller than the alpha{significance level}) then you reject the null hypothesis. In the opposite case you fail to reject the null hypothesis. This is for any hypothesis test-- not just one-tailed tests.

#6

If your null hypothesis is X ≤ Xh (X-hypothesized), then:

  • Find the t-critical value for your level of significance α. Remember the the t-critical value for a one-tailed t-test at α is the same as the t-critical value for a two-tailed t-test at 2α. This t-critical will be positive.
  • Compute your test statistic (for the mean it’ll be (X-bar – Xh)/sX, where sX is the sample std. dev. of X).
  • If the test statistic is greater than t-critical, reject the null hypothesis; if not, fail to reject the null.

If your null hypothesis is X ≥ Xh, then:

  • Find the t-critical value for your level of significance α. This t-critical will be negative.
  • Compute your test statistic,
  • If the test statistic is less than t-critical, reject the null hypothesis; if not, fail to reject the null
#7

Thanks. It’s helpful.

#8

You’re quite welcome. I’m glad that it was.

#9

Can you please explain this sentence: “…thicker tails mean more observations away from the center of the distribution (more outliers). Hence, hypothesis testing using the t-distribution makes it more difficult to reject the null relative to hypothesis testing using the z-distribution.” Does it mean that, with the same calculated test statistics value, t-test would result in a greater p-value, and thus it’s harder for it to be less than the significant level?

#10

Can you please explain this sentence: “…thicker tails mean more observations away from the center of the distribution (more outliers). Hence, hypothesis testing using the t-distribution makes it more difficult to reject the null relative to hypothesis testing using the z-distribution.” Does it mean that, with the same calculated test statistics value, t-test would result in a greater p-value, and thus it’s harder for it to be less than the significant level?

#11

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is this table 1 tailed or 2 tailed? i would guess since the problem is two tailed u would be given a 2 tailed table but i think this is the one tailed table… can someone help me?

#12

It’s a 1-tailed table. As an example, with df = 23 and going +2.069 standard deviations from the mean leaves 2.5% probability in the right hand tail. The 1/2 tail assumption should have been made clear. :roll_eyes:

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#13

So on this problem question:

For a sample size of 17, with a mean of 116.23 and a variance of 245.55, the width of a 90% confidence interval using the appropriate t-distribution is closest to:

So we simply know the DF is 16

What confuses me is:

  1. If it isn’t specified as one tail or two tail, does the default is two tail?
  2. If it’s two tail, do we check the " Level of significance for two tail table" at

a. check the value at 10% / 2 = 5% = 0.05 significance = 2.120

or

b. check the value at 10% = 0.1 significance = 1.746

  1. If it’s one tail, do we simply check the 10% value at one tail table? significance =1.337

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#14

Well, at least they specify how to read the table for 1 vs 2 tailed test. :unamused:

In this case, you want a 90% confidence interval around the sample mean, This means you have 45% above and below the sample mean with 10% left in the tails, or equivalently 5% in each tail.

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