For the hypothesis test H0: μ=10 against H1: μ>10 and variance known, calculate the P -value for

For the hypothesis test H0: μ=10 against H1: μ>10 and...

For the hypothesis test $H_{0}: \mu=10$ against $H_{1}: \mu>10$ and variance known, calculate the $P$ -value for each of the following test statistics.
(a) $z_{0}=2.05$
(b) $z_{0}=-1.84$
(c) $z_{0}=0.4$

Key Concepts

P-value Calculation

The p-value is the probability of observing a test statistic at least as extreme as the one measured, under the assumption that the null hypothesis is true. For a right-tailed test, this involves computing the area under the standard normal curve to the right of the observed z-score, which represents the likelihood of observing such a statistic by chance.

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It serves as the benchmark for calculating probabilities associated with Z-test statistics, where areas under the curve correspond to p-values.

Z-Test Statistic

The Z-test statistic is used when the population variance is known and the sampling distribution of the test statistic can be approximated by a normal distribution. It transforms the sampling data into a standardized score (z-score) that indicates how many standard deviations the sample data is from the hypothesized population mean.

One-Tailed Test

A one-tailed test examines whether a parameter is either greater than or less than a certain value, as specified by the alternative hypothesis. In a right-tailed test, the alternative hypothesis states that the parameter is greater than a specific value, and the p-value is computed as the probability of obtaining a value as extreme as or more extreme than the observed statistic in that direction.

Hypothesis Testing

Hypothesis testing is a statistical process used to decide whether there is enough evidence to reject a presumed state (the null hypothesis) in favor of an alternative hypothesis. It involves comparing observed data to what is expected under the null hypothesis in order to judge whether any deviation is statistically significant.

Transcript

00:01 Here we are given a hypothesis test for which the no hypothesis is that the mean is equal to 10, and the alternative hypothesis is that the mean is greater than 10.

00:11 And we are asked to calculate the p -value for several test statistics.

00:19 For part a, we are given the test statistic of 2 .05.

00:30 One thing we can note is that the alternative hypothesis is that the mean is greater than some value, which means that this is a one -sided or upper -tailed hypothesis test.

00:46 And so the p -value is the probability of getting a test statistic greater than z -0.

00:55 So visually, this means, let's suppose our z -0 is here.

01:01 And the alternative hypothesis is that the population mean is greater than some number.

01:09 Then it is an upper -tailed test, and the p -value is this area.

01:14 Which represents the probability of getting a test statistic at least as extreme as z -0.

01:28 And we can do this by finding the value of this or the value of this.

02:05 And this gives us a p -value of 0 .0 -02.

02:14 For part b, we're asked to find the p -value, but this time we're told the test statistic came out to minus 1 .84.

02:34 Now, just going back to the curve, this time we have a test statistic that is, let's say this is negative 1 .84.

02:45 However, based on our alternative hypothesis, this is an upper -tailed test...

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