A pharmaceutical manufacturer is concerned that the impurity...

A pharmaceutical manufacturer is concerned that the impurity concentration in pills should not exceed $3 \%$. It is known that from a particular production run impurity concentrations follow a normal distribution with a standard deviation of $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$. a. Test at the $5 \%$ level the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$. b. Find the $p$-value for this test. c. Suppose that the alternative hypothesis had been two-sided, rather than one-sided, with the null hypothesis $H_0: \mu=3$. State, without doing the calculations, whether the $p$-value of the test would be higher than, lower than, or the same as that found in part (b). Sketch a graph to illustrate your reasoning. d. In the context of this problem, explain why a onesided alternative hypothesis is more appropriate than a two-sided alternative.

A pharmaceutical manufacturer is concerned that the impurity concentration in pills should not exceed $3 \%$. It is known that from a particular production run impurity concentrations follow a normal distribution with a standard deviation of $0.4 \%$. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be $3.07 \%$.
a. Test at the $5 \%$ level the null hypothesis that the population mean impurity concentration is $3 \%$ against the alternative that it is more than $3 \%$.
b. Find the $p$-value for this test.
c. Suppose that the alternative hypothesis had been two-sided, rather than one-sided, with the null hypothesis $H_0: \mu=3$. State, without doing the calculations, whether the $p$-value of the test would be higher than, lower than, or the same as that found in part (b). Sketch a graph to illustrate your reasoning.
d. In the context of this problem, explain why a onesided alternative hypothesis is more appropriate than a two-sided alternative.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis, which is a statement of no effect or status quo, and an alternative hypothesis, which is contrary to the null. The test then determines, based on sample data and a chosen significance level, whether to reject the null hypothesis in favor of the alternative.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) typically states that the population parameter, such as a mean, is equal to a specified value. The alternative hypothesis (H1) claims that the parameter is different from that value. In the context of the problem, the null hypothesis asserts that the mean impurity concentration is 3%, while the one-sided alternative hypothesis claims it is greater than 3%. This formulation is crucial as it sets the framework for the statistical test and determines the direction of the critical region.

Significance Level (?)

The significance level, commonly 5% in many tests, is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error—rejecting a true null hypothesis. Setting this level establishes a cutoff point for the test statistic, beyond which the null hypothesis is considered unlikely given the sample data.

Z-Test and Standard Error

A Z-test is used when the population standard deviation is known and the sample size is sufficiently large, making the sampling distribution of the mean approximately normal. The standard error, calculated as the population standard deviation divided by the square root of the sample size, quantifies the spread of the sample mean around the true population mean. This forms the basis for constructing the Z-statistic, which is compared against critical values from the normal distribution.

p-value Interpretation

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, what was actually observed, assuming the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis. In one-sided tests, the p-value is computed for the specific tail of interest, while in two-sided tests, it takes into account both tails, typically resulting in a larger p-value for the same observed statistic.

One-sided vs. Two-sided Tests

Choosing between a one-sided and a two-sided test depends on the context of the hypothesis. A one-sided test is more appropriate when the research question or practical concern specifies a direction, such as ensuring that impurity does not exceed a maximum level. This allows for more statistical power in detecting an effect in that direction. In contrast, a two-sided test is used when deviations in both directions are of interest, thereby splitting the significance level between the two tails of the distribution.

Transcript

0:00 Hello.

00:01 So here we have our null hypothesis is that mu is equal to 3 % versus our alternative hypothesis that mu is greater than 3 % with our standard deviation.

00:11 Sigma is going to be equal to 0 .4%.

00:15 And our sample size, n is equal to 64 pills.

00:20 And our sample mean x bar is equal to 3 .07 % with our significance level alpha equal to 0 .0 .0 .0 .5%.

00:32 So then to test this, we do not reject our null hypothesis since we get that our test statistic, z is going to be equal to 3 .07 minus 3 divided by 0 .4, divided by the square root of 64.

00:49 That's going to be equal to 1 .4, which is going to be less than 1 .645.

00:56 So therefore we have that x, we have x bar is 3 .07, which is less than x subc bar, which is 3 .082.

01:05 So we can conclude that the population mean impurity concentration then is equal to 3%.

01:17 And then for part b, we have the p value for this test is going to be the probability that z sub p is greater than 1 .4.

01:27 0 .4.

01:29 So from our table, that's going to be 0 .0808.

01:34 And then we have, for the next part here, the p value is going to be the probability of obtaining a value of test statistics as extreme or as or more extreme than the actual value obtained when the null hypothesis is true.

01:52 So if we suppose that mu is equal to 3 % versus our alternative, the mu is not equal to 3%, then the p value for our two -tailed test is going to be, well, two times the probability that the absolute value of x bar minus mu not divided by sigma over the square root of n is going to be greater than z sub p over two, such that we have that the null hypothesis mu is equal to mu not...

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