The concentration of an active ingredient in the output of a...

The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst $A$ is used, the population mean concentration exceeds $65 \%$. The standard deviation is known to be $a=5 \%$ A sample of outputs from 30 independent experiments gives the average concentration of $x_{A}=64.5 \%$. (a) Does this sample information with an average concentration of $\bar{x}_{A}=64.5 \%$ provide disturbing information that perhaps $\mu_{A}$ is not $65 \%,$ but less than $65 \% ?$ Support your answer with a probability statement. (b) Suppose a similar experiment is done with the use of another catalyst, catalyst $B$. The standard deviation $a$ is still assumed to be $5 \%$ and $\bar{x}_{B}$ turns out to be $70 \%$. Comment on whether or not the sample information on catalyst $B$ seems to give strong information that suggests that $\mu_{B}$ is truly greater than $\mu_{A}$. Support your answer by computing $$P\left(\bar{X}_{B}-X_{A} \geq 5.5 \quad \mid \mu_{B}=\mu_{A}\right)$$. (c) Under the condition that $\mu_{A}=\mu_{B}=65 \%,$ give the approximate distribution of the following quantities (with mean and variance of each). Make use of the central limit theorem.

The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst $A$ is used, the population mean concentration exceeds $65 \%$. The standard deviation is known to be $a=5 \%$ A sample of outputs from 30 independent experiments gives the average concentration of $x_{A}=64.5 \%$.
(a) Does this sample information with an average concentration of $\bar{x}_{A}=64.5 \%$ provide disturbing information that perhaps $\mu_{A}$ is not $65 \%,$ but less than $65 \% ?$ Support your answer with a probability statement.
(b) Suppose a similar experiment is done with the use of another catalyst, catalyst $B$. The standard deviation $a$ is still assumed to be $5 \%$ and $\bar{x}_{B}$ turns out to be $70 \%$. Comment on whether or not the sample information on catalyst $B$ seems to give strong information that suggests that $\mu_{B}$ is truly greater than $\mu_{A}$. Support your answer by computing $$P\left(\bar{X}_{B}-X_{A} \geq 5.5 \quad \mid \mu_{B}=\mu_{A}\right)$$.
(c) Under the condition that $\mu_{A}=\mu_{B}=65 \%,$ give the approximate distribution of the following quantities (with mean and variance of each). Make use of the central limit theorem.

Key Concepts

Distribution of the Difference of Means

When comparing two independent sample means, the difference between them has its own sampling distribution. If the samples are large enough or the underlying distributions are normal, this difference is approximately normally distributed. The mean of the difference is the difference in the population means, and its variance is the sum of the variances of the individual sample means, assuming independence between samples.

Central Limit Theorem (CLT)

The Central Limit Theorem states that the distribution of the sample mean of a sufficiently large number of independent and identically distributed observations approaches a normal distribution, regardless of the shape of the underlying distribution. This theorem justifies the use of normal approximations in many statistical procedures, including the construction of confidence intervals and hypothesis testing for means.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of the sample mean computed from many independent samples taken from the same population. Its mean is equal to the population mean, and its standard deviation, known as the standard error, is the population standard deviation divided by the square root of the sample size. This concept is essential for estimating how much the sample mean is expected to vary from the true population mean.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves formulating a null hypothesis that represents a default or status quo condition, and an alternative hypothesis that represents a claim or effect to be tested. A test statistic is computed from the sample data, and its probability (p-value) under the null hypothesis is used to judge the strength of evidence against the null hypothesis.

P-value

The p-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic as extreme or more extreme than the one observed. It provides a measure of the evidence against the null hypothesis; a small p-value indicates that the observed data would be unlikely under the null hypothesis, leading to its rejection in favor of the alternative.

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