Questions: What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter: 11.9, 14.4, 11.6, 14.5, 15.6, 8.7, 12.5, 8.6, 11.5, 17.5 (a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test. It is necessary to check that the population is approximately normal because the sample size is small (n ≤ 30). (b) Following is a dotplot of the data. Is it reasonable to assume that the population is approximately normal? (c) Assume that the population standard deviation is σ=4. Perform a hypothesis test at the α=0.10 level using the critical value method with the table to determine whether you can conclude that the mean concentration of lead meets the EPA standard. What do you conclude? State the null and alternate hypotheses.

What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter: 11.9, 14.4, 11.6, 14.5, 15.6, 8.7, 12.5, 8.6, 11.5, 17.5 (a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test. It is necessary to check that the population is approximately normal because the sample size is small (n ≤ 30). (b) Following is a dotplot of the data. Is it reasonable to assume that the population is approximately normal? (c) Assume that the population standard deviation is σ=4. Perform a hypothesis test at the α=0.10 level using the critical value method with the table to determine whether you can conclude that the mean concentration of lead meets the EPA standard. What do you conclude? State the null and alternate hypotheses.
Transcript text: What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter: 11.9, 14.4, 11.6, 14.5, 15.6, 8.7, 12.5, 8.6, 11.5, 17.5 (a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test. It is necessary to check that the population is approximately normal because the sample size is small ( \(n \leq 30\) ). (b) Following is a dotplot of the data. Is it reasonable to assume that the population is approximately normal? (c) Assume that the population standard deviation is \(\sigma=4\). Perform a hypothesis test at the \(\alpha=0.10\) level using the critical value method with the table to determine whether you can conclude that the mean concentration of lead meets the EPA standard. What do you conclude? State the null and alternate hypotheses.
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Solution

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Solution Steps

Step 1: Reason for checking for population normality

It is necessary to check that the population is approximately normal before performing a hypothesis test because the sample size is small (n=10, which is ≤ 30). When the sample size is small, the Central Limit Theorem doesn't guarantee that the sampling distribution of the mean will be approximately normal, so we need to check the population distribution.

Step 2: Dotplot analysis

Based on the dotplot, it seems reasonable to assume that the population is approximately normal. The dotplot does not show strong skewness or outliers. While the sample size is small, there is no evidence from the dotplot to suggest non-normality.

Step 3: Hypothesis Test setup

Null Hypothesis (H₀): μ ≥ 15 (The mean lead concentration is greater than or equal to 15 micrograms per liter) Alternative Hypothesis (H₁): μ < 15 (The mean lead concentration is less than 15 micrograms per liter) This is a left-tailed test because we are investigating if the mean lead concentration is _less than_ the EPA standard.

Final Answer:

  1. Normality is checked due to small sample size.
  2. Dotplot appears approximately normal.
  3. Null Hypothesis: μ ≥ 15, Alternative Hypothesis: μ < 15, Left-tailed test.
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