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State a quantity or proportion Probability of Observing a Quantity or Proportion Quantity Excel functions standard deviation of quantities stdev(data) area to the left of t-score t.dist(t-score, degrees of freedom, TRUE) area to the left of z-score = norm.s.dist(z-score, TRUE) Proportion Calculate the score t-score stated quantity sample mean standard deviation Calculate the score z-score stated proportion sample proportion standard deviation Find the area to the left of the score t The shaded area is the probability of observing a quantity or proportion less than or equal to the stated quantity or proportion. State the hypothesis Hypothesis Test for a Single Proportion Comparing a sample proportion to a fixed quantity H0 : population proportion k standard error HA proportion > k HA proportion < k HA ≠ k 1 k N Calculate the score z-score Confidence Interval p critical value sample proportion k standard error p 1 p N Find the p-value (standard normal distribution) p-value = 1 – norm.s.dist(z-score, TRUE) p-value = 2 x (1 – norm.s.dist(abs(z-score), TRUE)) p-value = norm.s.dist(z-score, TRUE) Excel functions area left of the score = norm.s.dist(z-score, TRUE) z-score = norm.s.inv(area left of the score) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis. State the hypothesis Hypothesis Test for Comparing Proportions Comparing two sample proportions HA > HA < HA ≠ H0 : population proportion 1 = population proportion 2 standard error p1 1 p1 p2 1 p2 N1 N2 Calculate the score z-score sample proportion 1 sample proportion 2 standard error Confidence Interval p1 p2 critical value p1 1 p1 p2 1 p2 N1 N2 Find the p-value (standard normal distribution) p-value = 1 – norm.s.dist(z-score, TRUE) p-value = 2 x (1 – norm.s.dist(abs(z-score), TRUE)) p-value = norm.s.dist(z-score, TRUE) Excel functions area left of the score = norm.s.dist(z-score, TRUE) z-score = norm.s.inv(area left of the score) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis. State the hypothesis Hypothesis Test for a Single Mean Comparing a sample mean to a fixed quantity H0 : population mean = k HA mean > k HA mean < k s2 N degrees of freedom N 1 HA ≠ standard error Calculate the score t-score sample mean k standard error Confidence Interval s2 N x critical value Find the p-value (t distribution) p-value = 1 – t.dist(t-score, df, TRUE) p-value = 2 x (1 – t.dist(abs(t-score), df, TRUE)) t p-value = t.dist(t-score, df, TRUE) Excel functions area left of the score = t.dist(t-score, degrees of freedom, TRUE) t-score = t.inv(area left of the score, degrees of freedom) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis. State the hypothesis Hypothesis Test for Comparing Means Comparing two sample means H0 : population mean 1 = population mean 2 standard error HA > HA < HA ≠ s12 s22 N1 N2 Calculate the score degrees of freedom N1 N2 2 t-score sample mean 1 sample mean 2 standard error Confidence Interval x1 x2 critical value s12 s22 N1 N2 Find the p-value (t distribution) p-value = 1 – t.dist(t-score, df, TRUE) p-value = 2 x (1 – t.dist(abs(t-score), df, TRUE)) t Excel functions area left of the score = t.dist(t-score, degrees of freedom, TRUE) t-score = t.inv(area left of the score, degrees of freedom) area to the nearest tail (two-tailed test) t.test(data set 1, data set 2, 2, 3) area to the nearest tail (one-tailed test) t.test(data set 1, data set 2, 1, 3) p-value = t.dist(t-score, df, TRUE) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis. State the hypothesis Hypothesis Test for a Single Variance Comparing a variance to a fixed quantity H0 : population variance = k HA variance > k HA variance < k HA ≠ degrees of freedom N 1 Calculate the score 2 -score N 1 s2 k Find the p-value (Chi-squared distribution) p-value = 1 – chi.dist(x-score, df, TRUE) Excel functions area left of the score = chisq.dist(score, degrees of freedom, TRUE) 2 -score = chisq.inv(area left of the score, degrees of freedom) p-value = 2 x (1 – chi.dist(x-score, df, TRUE)) or 2 x chi.dist(x-score, df, TRUE), whichever is less p-value = chi.dist(x-score, df, TRUE) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis. State the hypothesis Hypothesis Test for Comparing Variances Comparing two variances H0 : population variance 1 = population variance 2 HA larger > smaller HA larger < smaller One of the sample variances will be larger than the other. degrees of freedom larger sample variance = Nlarger variance 1 HA ≠ Calculate the score degrees of freedom smaller sample variance Nsmaller variance 1 F-score 2 slarger variance 2 ssmaller variance Find the p-value (F distribution) p-value = 1 – f.dist(f-score, df, df, TRUE) Excel functions area left of the score = f.dist(f-score, df for larger variance, df for smaller variance, TRUE) F-score = f.inv(area left of the score, df for larger variance, df for smaller variance) p-value = 2 x (1 – f.dist(f-score, df, df, TRUE)) p-value = f.dist(f-score, df, df, TRUE) State the conclusion p-value approach: The probability of the null hypothesis being true is the p-value. Significance approach: If p-value < significance level, reject the null hypothesis.
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