# doc-cache created by Octave 5.2.0
# name: cache
# type: cell
# rows: 3
# columns: 25
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
anova


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1011
 -- [PVAL, F, DF_B, DF_W] = anova (Y, G)
     Perform a one-way analysis of variance (ANOVA).

     The goal is to test whether the population means of data taken from
     K different groups are all equal.

     Data may be given in a single vector Y with groups specified by a
     corresponding vector of group labels G (e.g., numbers from 1 to K).
     This is the general form which does not impose any restriction on
     the number of data in each group or the group labels.

     If Y is a matrix and G is omitted, each column of Y is treated as a
     group.  This form is only appropriate for balanced ANOVA in which
     the numbers of samples from each group are all equal.

     Under the null of constant means, the statistic F follows an F
     distribution with DF_B and DF_W degrees of freedom.

     The p-value (1 minus the CDF of this distribution at F) is returned
     in PVAL.

     If no output argument is given, the standard one-way ANOVA table is
     printed.

     See also: manova.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Perform a one-way analysis of variance (ANOVA).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
bartlett_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 472
 -- [PVAL, CHISQ, DF] = bartlett_test (X1, ...)
     Perform a Bartlett test for the homogeneity of variances in the
     data vectors X1, X2, ..., XK, where K > 1.

     Under the null of equal variances, the test statistic CHISQ
     approximately follows a chi-square distribution with DF degrees of
     freedom.

     The p-value (1 minus the CDF of this distribution at CHISQ) is
     returned in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a Bartlett test for the homogeneity of variances in the data
vectors ...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 26
chisquare_test_homogeneity


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 592
 -- [PVAL, CHISQ, DF] = chisquare_test_homogeneity (X, Y, C)
     Given two samples X and Y, perform a chisquare test for homogeneity
     of the null hypothesis that X and Y come from the same
     distribution, based on the partition induced by the (strictly
     increasing) entries of C.

     For large samples, the test statistic CHISQ approximately follows a
     chisquare distribution with DF = 'length (C)' degrees of freedom.

     The p-value (1 minus the CDF of this distribution at CHISQ) is
     returned in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Given two samples X and Y, perform a chisquare test for homogeneity of
the nu...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 27
chisquare_test_independence


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 442
 -- [PVAL, CHISQ, DF] = chisquare_test_independence (X)
     Perform a chi-square test for independence based on the contingency
     table X.

     Under the null hypothesis of independence, CHISQ approximately has
     a chi-square distribution with DF degrees of freedom.

     The p-value (1 minus the CDF of this distribution at chisq) of the
     test is returned in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 76
Perform a chi-square test for independence based on the contingency
table X.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
cor_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1364
 -- cor_test (X, Y, ALT, METHOD)
     Test whether two samples X and Y come from uncorrelated
     populations.

     The optional argument string ALT describes the alternative
     hypothesis, and can be "!=" or "<>" (nonzero), ">" (greater than
     0), or "<" (less than 0).  The default is the two-sided case.

     The optional argument string METHOD specifies which correlation
     coefficient to use for testing.  If METHOD is "pearson" (default),
     the (usual) Pearson's product moment correlation coefficient is
     used.  In this case, the data should come from a bivariate normal
     distribution.  Otherwise, the other two methods offer nonparametric
     alternatives.  If METHOD is "kendall", then Kendall's rank
     correlation tau is used.  If METHOD is "spearman", then Spearman's
     rank correlation rho is used.  Only the first character is
     necessary.

     The output is a structure with the following elements:

     PVAL
          The p-value of the test.

     STAT
          The value of the test statistic.

     DIST
          The distribution of the test statistic.

     PARAMS
          The parameters of the null distribution of the test statistic.

     ALTERNATIVE
          The alternative hypothesis.

     METHOD
          The method used for testing.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 68
Test whether two samples X and Y come from uncorrelated populations.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 17
f_test_regression


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 487
 -- [PVAL, F, DF_NUM, DF_DEN] = f_test_regression (Y, X, RR, R)
     Perform an F test for the null hypothesis rr * b = r in a classical
     normal regression model y = X * b + e.

     Under the null, the test statistic F follows an F distribution with
     DF_NUM and DF_DEN degrees of freedom.

     The p-value (1 minus the CDF of this distribution at F) is returned
     in PVAL.

     If not given explicitly, R = 0.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform an F test for the null hypothesis rr * b = r in a classical
normal re...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 14
hotelling_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 551
 -- [PVAL, TSQ] = hotelling_test (X, M)
     For a sample X from a multivariate normal distribution with unknown
     mean and covariance matrix, test the null hypothesis that 'mean (X)
     == M'.

     Hotelling's T^2 is returned in TSQ.  Under the null, (n-p) T^2 /
     (p(n-1)) has an F distribution with p and n-p degrees of freedom,
     where n and p are the numbers of samples and variables,
     respectively.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For a sample X from a multivariate normal distribution with unknown mean
and ...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 16
hotelling_test_2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 667
 -- [PVAL, TSQ] = hotelling_test_2 (X, Y)
     For two samples X from multivariate normal distributions with the
     same number of variables (columns), unknown means and unknown equal
     covariance matrices, test the null hypothesis 'mean (X) == mean
     (Y)'.

     Hotelling's two-sample T^2 is returned in TSQ.  Under the null,

          (n_x+n_y-p-1) T^2 / (p(n_x+n_y-2))

     has an F distribution with p and n_x+n_y-p-1 degrees of freedom,
     where n_x and n_y are the sample sizes and p is the number of
     variables.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples X from multivariate normal distributions with the same
number...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 23
kolmogorov_smirnov_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1323
 -- [PVAL, KS] = kolmogorov_smirnov_test (X, DIST, PARAMS, ALT)
     Perform a Kolmogorov-Smirnov test of the null hypothesis that the
     sample X comes from the (continuous) distribution DIST.

     if F and G are the CDFs corresponding to the sample and dist,
     respectively, then the null is that F == G.

     The optional argument PARAMS contains a list of parameters of DIST.
     For example, to test whether a sample X comes from a uniform
     distribution on [2,4], use

          kolmogorov_smirnov_test (x, "unif", 2, 4)

     DIST can be any string for which a function DISTCDF that calculates
     the CDF of distribution DIST exists.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative F != G.  In this case, the test
     statistic KS follows a two-sided Kolmogorov-Smirnov distribution.
     If ALT is ">", the one-sided alternative F > G is considered.
     Similarly for "<", the one-sided alternative F > G is considered.
     In this case, the test statistic KS has a one-sided
     Kolmogorov-Smirnov distribution.  The default is the two-sided
     case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a Kolmogorov-Smirnov test of the null hypothesis that the sample
X co...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 25
kolmogorov_smirnov_test_2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1145
 -- [PVAL, KS, D] = kolmogorov_smirnov_test_2 (X, Y, ALT)
     Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis
     that the samples X and Y come from the same (continuous)
     distribution.

     If F and G are the CDFs corresponding to the X and Y samples,
     respectively, then the null is that F == G.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative F != G.  In this case, the test
     statistic KS follows a two-sided Kolmogorov-Smirnov distribution.
     If ALT is ">", the one-sided alternative F > G is considered.
     Similarly for "<", the one-sided alternative F < G is considered.
     In this case, the test statistic KS has a one-sided
     Kolmogorov-Smirnov distribution.  The default is the two-sided
     case.

     The p-value of the test is returned in PVAL.

     The third returned value, D, is the test statistic, the maximum
     vertical distance between the two cumulative distribution
     functions.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis that
the sa...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 19
kruskal_wallis_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1114
 -- [PVAL, K, DF] = kruskal_wallis_test (X1, ...)
     Perform a Kruskal-Wallis one-factor analysis of variance.

     Suppose a variable is observed for K > 1 different groups, and let
     X1, ..., XK be the corresponding data vectors.

     Under the null hypothesis that the ranks in the pooled sample are
     not affected by the group memberships, the test statistic K is
     approximately chi-square with DF = K - 1 degrees of freedom.

     If the data contains ties (some value appears more than once) K is
     divided by

     1 - SUM_TIES / (N^3 - N)

     where SUM_TIES is the sum of T^2 - T over each group of ties where
     T is the number of ties in the group and N is the total number of
     values in the input data.  For more info on this adjustment see
     William H. Kruskal and W. Allen Wallis, 'Use of Ranks in
     One-Criterion Variance Analysis', Journal of the American
     Statistical Association, Vol.  47, No.  260 (Dec 1952).

     The p-value (1 minus the CDF of this distribution at K) is returned
     in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Perform a Kruskal-Wallis one-factor analysis of variance.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
manova


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 652
 -- manova (X, G)
     Perform a one-way multivariate analysis of variance (MANOVA).

     The goal is to test whether the p-dimensional population means of
     data taken from K different groups are all equal.  All data are
     assumed drawn independently from p-dimensional normal distributions
     with the same covariance matrix.

     The data matrix is given by X.  As usual, rows are observations and
     columns are variables.  The vector G specifies the corresponding
     group labels (e.g., numbers from 1 to K).

     The LR test statistic (Wilks' Lambda) and approximate p-values are
     computed and displayed.

     See also: anova.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 61
Perform a one-way multivariate analysis of variance (MANOVA).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 12
mcnemar_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 535
 -- [PVAL, CHISQ, DF] = mcnemar_test (X)
     For a square contingency table X of data cross-classified on the
     row and column variables, McNemar's test can be used for testing
     the null hypothesis of symmetry of the classification
     probabilities.

     Under the null, CHISQ is approximately distributed as chisquare
     with DF degrees of freedom.

     The p-value (1 minus the CDF of this distribution at CHISQ) is
     returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For a square contingency table X of data cross-classified on the row and
colu...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
prop_test_2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 843
 -- [PVAL, Z] = prop_test_2 (X1, N1, X2, N2, ALT)
     If X1 and N1 are the counts of successes and trials in one sample,
     and X2 and N2 those in a second one, test the null hypothesis that
     the success probabilities P1 and P2 are the same.

     Under the null, the test statistic Z approximately follows a
     standard normal distribution.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative P1 != P2.  If ALT is ">", the
     one-sided alternative P1 > P2 is used.  Similarly for "<", the
     one-sided alternative P1 < P2 is used.  The default is the
     two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
If X1 and N1 are the counts of successes and trials in one sample, and
X2 and...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
run_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 336
 -- [PVAL, CHISQ] = run_test (X)
     Perform a chi-square test with 6 degrees of freedom based on the
     upward runs in the columns of X.

     'run_test' can be used to decide whether X contains independent
     data.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value is displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a chi-square test with 6 degrees of freedom based on the upward
runs ...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
sign_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 928
 -- [PVAL, B, N] = sign_test (X, Y, ALT)
     For two matched-pair samples X and Y, perform a sign test of the
     null hypothesis PROB (X > Y) == PROB (X < Y) == 1/2.

     Under the null, the test statistic B roughly follows a binomial
     distribution with parameters 'N = sum (X != Y)' and P = 1/2.

     With the optional argument 'alt', the alternative of interest can
     be selected.  If ALT is "!=" or "<>", the null hypothesis is tested
     against the two-sided alternative PROB (X < Y) != 1/2.  If ALT is
     ">", the one-sided alternative PROB (X > Y) > 1/2 ("x is
     stochastically greater than y") is considered.  Similarly for "<",
     the one-sided alternative PROB (X > Y) < 1/2 ("x is stochastically
     less than y") is considered.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two matched-pair samples X and Y, perform a sign test of the null
hypothe...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
t_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 830
 -- [PVAL, T, DF] = t_test (X, M, ALT)
     For a sample X from a normal distribution with unknown mean and
     variance, perform a t-test of the null hypothesis 'mean (X) == M'.

     Under the null, the test statistic T follows a Student distribution
     with 'DF = length (X) - 1' degrees of freedom.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'mean (X) != M'.  If ALT is ">",
     the one-sided alternative 'mean (X) > M' is considered.  Similarly
     for "<", the one-sided alternative 'mean (X) < M' is considered.
     The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For a sample X from a normal distribution with unknown mean and
variance, per...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
t_test_2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 861
 -- [PVAL, T, DF] = t_test_2 (X, Y, ALT)
     For two samples x and y from normal distributions with unknown
     means and unknown equal variances, perform a two-sample t-test of
     the null hypothesis of equal means.

     Under the null, the test statistic T follows a Student distribution
     with DF degrees of freedom.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'mean (X) != mean (Y)'.  If ALT
     is ">", the one-sided alternative 'mean (X) > mean (Y)' is used.
     Similarly for "<", the one-sided alternative 'mean (X) < mean (Y)'
     is used.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples x and y from normal distributions with unknown means and
unkn...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 17
t_test_regression


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 833
 -- [PVAL, T, DF] = t_test_regression (Y, X, RR, R, ALT)
     Perform a t test for the null hypothesis 'RR * B = R' in a
     classical normal regression model 'Y = X * B + E'.

     Under the null, the test statistic T follows a T distribution with
     DF degrees of freedom.

     If R is omitted, a value of 0 is assumed.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'RR * B != R'.  If ALT is ">",
     the one-sided alternative 'RR * B > R' is used.  Similarly for "<",
     the one-sided alternative 'RR * B < R' is used.  The default is the
     two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a t test for the null hypothesis 'RR * B = R' in a classical
normal r...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
u_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 869
 -- [PVAL, Z] = u_test (X, Y, ALT)
     For two samples X and Y, perform a Mann-Whitney U-test of the null
     hypothesis PROB (X > Y) == 1/2 == PROB (X < Y).

     Under the null, the test statistic Z approximately follows a
     standard normal distribution.  Note that this test is equivalent to
     the Wilcoxon rank-sum test.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative PROB (X > Y) != 1/2.  If ALT is
     ">", the one-sided alternative PROB (X > Y) > 1/2 is considered.
     Similarly for "<", the one-sided alternative PROB (X > Y) < 1/2 is
     considered.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples X and Y, perform a Mann-Whitney U-test of the null
hypothesis...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
var_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 865
 -- [PVAL, F, DF_NUM, DF_DEN] = var_test (X, Y, ALT)
     For two samples X and Y from normal distributions with unknown
     means and unknown variances, perform an F-test of the null
     hypothesis of equal variances.

     Under the null, the test statistic F follows an F-distribution with
     DF_NUM and DF_DEN degrees of freedom.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'var (X) != var (Y)'.  If ALT is
     ">", the one-sided alternative 'var (X) > var (Y)' is used.
     Similarly for "<", the one-sided alternative 'var (X) > var (Y)' is
     used.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples X and Y from normal distributions with unknown means and
unkn...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
welch_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 876
 -- [PVAL, T, DF] = welch_test (X, Y, ALT)
     For two samples X and Y from normal distributions with unknown
     means and unknown and not necessarily equal variances, perform a
     Welch test of the null hypothesis of equal means.

     Under the null, the test statistic T approximately follows a
     Student distribution with DF degrees of freedom.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'mean (X) != M'.  If ALT is ">",
     the one-sided alternative mean(x) > M is considered.  Similarly for
     "<", the one-sided alternative mean(x) < M is considered.  The
     default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples X and Y from normal distributions with unknown means and
unkn...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
wilcoxon_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 932
 -- [PVAL, Z] = wilcoxon_test (X, Y, ALT)
     For two matched-pair sample vectors X and Y, perform a Wilcoxon
     signed-rank test of the null hypothesis PROB (X > Y) == 1/2.

     Under the null, the test statistic Z approximately follows a
     standard normal distribution when N > 25.

     *Caution:* This function assumes a normal distribution for Z and
     thus is invalid for N <= 25.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative PROB (X > Y) != 1/2.  If alt is
     ">", the one-sided alternative PROB (X > Y) > 1/2 is considered.
     Similarly for "<", the one-sided alternative PROB (X > Y) < 1/2 is
     considered.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two matched-pair sample vectors X and Y, perform a Wilcoxon
signed-rank t...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
z_test


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 831
 -- [PVAL, Z] = z_test (X, M, V, ALT)
     Perform a Z-test of the null hypothesis 'mean (X) == M' for a
     sample X from a normal distribution with unknown mean and known
     variance V.

     Under the null, the test statistic Z follows a standard normal
     distribution.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'mean (X) != M'.  If ALT is ">",
     the one-sided alternative 'mean (X) > M' is considered.  Similarly
     for "<", the one-sided alternative 'mean (X) < M' is considered.
     The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed along with some information.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Perform a Z-test of the null hypothesis 'mean (X) == M' for a sample X
from a...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
z_test_2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 864
 -- [PVAL, Z] = z_test_2 (X, Y, V_X, V_Y, ALT)
     For two samples X and Y from normal distributions with unknown
     means and known variances V_X and V_Y, perform a Z-test of the
     hypothesis of equal means.

     Under the null, the test statistic Z follows a standard normal
     distribution.

     With the optional argument string ALT, the alternative of interest
     can be selected.  If ALT is "!=" or "<>", the null is tested
     against the two-sided alternative 'mean (X) != mean (Y)'.  If alt
     is ">", the one-sided alternative 'mean (X) > mean (Y)' is used.
     Similarly for "<", the one-sided alternative 'mean (X) < mean (Y)'
     is used.  The default is the two-sided case.

     The p-value of the test is returned in PVAL.

     If no output argument is given, the p-value of the test is
     displayed along with some information.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For two samples X and Y from normal distributions with unknown means and
know...





