Question

​Cadmium, a heavy​ metal, is toxic to animals.​ Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. A government set a safety limit for cadmium in dry vegetables at 0.5 parts per million​ (ppm). The cadmium levels in a random sample of one species of edible mushroom are in the accompanying data set. At the 10​% significance​ level, do the data provide sufficient evidence to conclude that the mean cadmium level in this species of mushroom is greater than the​ government's recommended limit of 0.5 ​ppm? Assume that the population standard deviation of cadmium levels in this species of mushroom is 0.38 ppm. Preliminary data analyses indicate that applying the​ z-test is reasonable.​ (Note: The sum of the data is 7.38 ​ppm.)

Answer 1

A one-sample z-test is used to determine if the mean cadmium level in mushrooms exceeds the safety limit of 0.5 ppm, given the known standard deviation. The null hypothesis states the mean level is 0.5 ppm or less, and it is rejected if the test statistic exceeds the critical value at a 10% significance level.

Step-by-step explanation:

When conducting a hypothesis test to determine if the mean cadmium level in a species of mushroom is greater than the government safety limit of 0.5 parts per million (ppm), we would use a one-sample z-test given that the population standard deviation is known (0.38 ppm).

The null hypothesis (H0) states that the mean cadmium level is 0.5 ppm or less, and the alternative hypothesis (Ha) states that the mean is greater than 0.5 ppm. To perform this test, we need the sample mean, which is calculated from the sum of the sample data provided (7.38 ppm) divided by the sample size.

We would then calculate the test statistic using the formula: (sample mean - population mean) / (population standard deviation / sqrt(sample size)). With the test statistic calculated and the significance level set at 10%, we would compare the test statistic to the critical value from the z-distribution.

If the test statistic is greater than the critical value, we reject the null hypothesis, suggesting that the cadmium levels in the mushrooms are higher than the safety limit.