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Results tagged with hypothesis-testing
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user 11667
This tag is for questions on hypothesis testing in statistics, including questions about constructing or setting up a test, selecting an appropriate test for a particular application, and computing test statistics.
0
votes
Neyman Pearson Lemma Question
You reject $H_0$ iff
$$
\left( \frac 1 {\theta_1} - \frac 1 {\theta_0} \right) \sum_i x_i \le \text{something}.
$$
Multiplying both sides by the reciprocal of $\displaystyle\left( \frac 1 {\theta_1} - …
1
vote
Accepted
Find the form of the most powerful critical region of size $\alpha$.
My first thought was that where you wrote $\displaystyle\prod_{x=1}^n \theta_0(1-\theta_0)^{x-1}$ and $\displaystyle \sum_{x=1}^n x,$ you need $\displaystyle \prod_{i=1}^n \theta_0(1-\theta_0)^{x_i-1} …
2
votes
Accepted
What is the significance of the measures of variation in hypothesis testing?
I shall assume you intended no pun and I won't even mention it.
If the observed mean is far from the hypothetical mean by comparison to what you would expect given its normal range of variation, then …
1
vote
Accepted
P-values : Hypothesis test
If "mean" is taken to be the mean of $15$ differences between the $15$ measured cut-on voltages and the hypothetical average of $0.6$, then my software agrees with the output in the posted question ab …
0
votes
Accepted
prove that test statistic follows normal distribution
The central limit theorem does not bear upon this, because the random variables you're starting with are already normally distributed. A linear combination of independent normally distributed random v …
1
vote
Accepted
Hypothesis testing
You have the right p-value, but you did not understand the software. The p-value that you say you got from R is for a two-sided alternative hypothesis that says $\mu\ne42$ instead of $\mu<42$.
Your …
1
vote
what should the null hypothesis be
As stated, the problem doesn't make a lot of sense as a hypothesis test. That this group of 50 scored higher than average is an empirical observation, so there's no occasion for statistical inference …
1
vote
The difference between a Z-Score and a Z-statistic? Why do we divide by $\sqrt{n}$ for the l...
A z-score is what you get when you subtract its expected value from a random variable and then divide by its standard deviation. Thus if $\operatorname E(X) = \mu$ and $\operatorname{s.d.}(X) = \sigma …
0
votes
Neyman–Pearson Lemma
Randomization in this context never gets you a better test than what you would have without randomization. With discrete distributions, the distribution of the p-value, assuming the null hypothesis is …
0
votes
$\chi^2$ distribution: hypothesis testing
The sum of independent chi-square random variables is itself a chi-square random variable. So
$$
X_1 + \cdots + X_{10} \sim \chi^2_{10k}\,. \tag 1
$$
Since this has expected value $10k$, the sample me …
1
vote
chi-square distribution and p-value
The only mistake I find in your posting is your assertion that $63.691<62.5.$
Since the value of the test statistic is less than the critical value, the null hypothesis is not rejected.
Likewise, si …
1
vote
Simple exercise on t-Student test
The smallness of the sample is precisely the reason why it's important to use the t-distribution rather than the normal distribution, in this case with $3$ degrees of freedom.
$\require{cancel}$
\beg …
1
vote
Hypothesis Testing a normally distributed random variable
You have $Y\sim N(\mu,\sigma^2)$ and the null and alternative hypotheses say $(\mu,\sigma) = (\mu_i,\sigma_i^2)$ for $i=1,2$ respectively.
Let $y$ be the observed value of $Y.$ The likelihood functio …
2
votes
Accepted
Determine the optimal critical region for a Poisson distribution
You have made use of an unstated assumption that $\lambda_0>\lambda_1.$
You concluded that the rejection criterion is that $\sum_{i=1}^n X_i < c$ where $c$ is the critical value.
If the level of the …
0
votes
Approximation of region of rejection
If the null hypothesis is true, then $X_1,\ldots,X_n\sim\mathrm{i.i.d.}\operatorname{Uniform}(0,1),$ so
$$
\text{for } y\ge0, \quad\Pr(-2\log X_1 \ge y) = \Pr(X_1\le e^{-y/2}) = e^{-y/2}.
$$
Therefore …