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Lab 4: What is a t-test?

Lab 4: What is a t-test?. Something British mothers use to see if the new girlfriend is significantly better than the old one? . The t Distribution. We want to compute a confidence interval & test a hypothesis for an unknown population mean µ

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Lab 4: What is a t-test?

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  1. Lab 4: What is a t-test? Something British mothers use to see if the new girlfriend is significantly better than the old one? 

  2. The t Distribution • We want to compute a confidence interval & test a hypothesis for an unknown population mean µ • To use the Z distribution we must know the population standard deviation σ • In most real life situations, we don’t know the true population standard deviation • In this case we can use the t distribution instead of the Z distribution to calculate confidence intervals & test hypotheses It seems the more we learn the less we know!

  3. T Distribution: Unimodal & symmetric around zero Use when population µ & σ are both UNKNOWN Assumes variable of interest is normally distributed Using sample S.D. introduces more sampling variability Heavier tails (n-1) degrees of freedom Z Distribution (CLT): Unimodal & symmetric around zero Use when population µ is UNKNOWN but σ is KNOWN CLT allows us to say the sampling distribution of the mean is approx. normal as n gets large, even when the underlying variable of interest is not normally distributed Smaller tails T vs. Z Distribution: Who’s who?

  4. T vs. Z Distribution: Who’s who?

  5. The t Distribution • Assumptions: • The variable (X) is normally distributed • Random sample of size n from the underlying population • Very similar to Z distribution as sample size gets “large” (30+) • (n-1) degrees of freedom

  6. Degrees of Freedom: (n-1) • The “Currency” of statistics - you earn a degree of freedom for every data point you collect, and you spend a degree of freedom for each parameter you estimate. Since you usually need to spend 1 just to calculate the mean, you then are left with n-1 (total data points "n" - 1 spent on calculating the mean). (Reference: http://www.isixsigma.com/dictionary) • A general rule is that the degrees of freedom decrease when we have to estimate more parameters. • Before you can compute the standard deviation, you have to first estimate a mean. • This causes you to lose a degree of freedom (Reference: http://www.childrens-mercy.org/stats/ask/df.asp) Two statistics are in a bar, talking and drinking. One statistic turns to the other and says "So how are you finding married life?" The other statistic responds, "It's okay, but you lose a degree of freedom."

  7. STATA & the one sample t-test

  8. STATA Options • Get t critical value: display invttail(df,p) • Used for CI & Hypothesis Tests • Get p-value: display ttail(df,t) (one-sided) display tprob(df,t) (two-sided) • Run t-test from data summary: • Useful for summary homework problems ttesti n x_bar s µ • Run t-test on actual data: • Useful in real-life research ttest varname= µ

  9. STATA: obtaining the critical value • Example: Concentration of benzene in cigars • Hypothesis Test: 2-sided test • Null Hypothesis: μ=81 μg/g vs. Alternative Hypothesis: μ≠81 μg/g • Standard deviation is unknown • α= 0.05 (two-sided test) • Data: • Sample mean= 151 μg/g • Sample Standard deviation, s=9 μg/g • d.f. = n-1 = 7-1 = 6 • The STATA command: invttail(df,p) where df is the degrees of freedom and p is a number between 0 and 1. • display invttail(6,0.025) 2.4469118 • This means that if T statistic is above 2.447 or below –2.447, then we would reject the null hypothesis at the 5% alpha level. Since the observed value of the statistic T is 20.6, we reject the null hypothesis. t=

  10. Note of Confusion! • Note! • invnorm(p) returns the inverse cumulative standard normal distribution [i.e. returns z which satisfies P(Z ≤ z)=p] • invttail(df,p) returns the inverse REVERSE cumulative Student's t distribution [i.e. returns t which satisfies P(T ≥ t)=p)] • So instead of using invttail(6,0.975) we should use invttail(6,0.025)

  11. STATA: obtaining the p-value • Use: ttail(df,t) (one-sided) or tprob(df,t) (two-sided) display ttail(6,20.6) 4.257e-07 • This gives you P(T ≥ 20.6). To obtain the p-value for this two-sided test, we have p=P(|T| ≥ 20.6) = P(T ≥ 20.6 or T ≤ -20.6)=2*P(T ≥ 20.6)= 8.513e-07. • While tprob will give you P(|T| ≥ t) directly: display tprob(6,20.6) 8.513e-07

  12. STATA: running one-sample t-test from summary statistics • ttesti n x_bar s µ • Null Hypothesis: μ=81 μg/g vs. Alternative Hypothesis: μ≠81 μg/g Data: Sample mean= 151 μg/g (x_bar) Sample Standard deviation, s=9 μg/g d.f. = n-1 = 7-1 = 6 ttesti 7 151 9 81

  13. STATA Output • One-sample t test • ------------------------------------------------------------------------------ • | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] • ---------+-------------------------------------------------------------------- • x | 7 151 3.40168 9 142.6764 159.3236 • ------------------------------------------------------------------------------ • Degrees of freedom: 6 • Ho: mean(x) = 81 • Ha: mean < 81 Ha: mean != 81 Ha: mean > 81 • t = 20.5781 t = 20.5781 t = 20.5781 • P < t = 1.0000 P > |t| = 0.0000 P > t = 0.0000

  14. STATA: running one-sample t-test on data • Open lowbwt.dta contained on the disk in your book. If you wish to test a hypothesis regarding the population mean of the gestation age of low birth weight infants (for example: you might hypothesize that low birth infants have gestation ages greater than 28 weeks). To test this one-sided hypothesis: H0: mean <= 28 H1: mean > 28 Alpha-level = 0.05 You would use the following STATA command: ttest gestage = 28

  15. STATA Output • One-sample t test • ------------------------------------------------------------------------------ • Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] • ---------+-------------------------------------------------------------------- • gestage | 100 28.89 .253419 2.53419 28.38716 29.39284 • ------------------------------------------------------------------------------ • Degrees of freedom: 99 • Ho: mean(gestage) = 28 • Ha: mean < 28 Ha: mean != 28 Ha: mean > 28 • t = 3.5120 t = 3.5120 t = 3.5120 • P < t = 0.9997 P > |t| = 0.0007 P > t = 0.0003 • STATA writes out two-sided and one-sided hypotheses. In this case, we would be employing the one on the right (Ha: mean>28). Since the p-value is 0.0003 which is less than our alpha-level of 0.05, we would reject the null hypothesis and conclude that the mean gestation age is not less than 28 weeks.

  16. STATA: two sample t-test & paired t-test Next Week…

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