Question

1. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 45.29 ng/mL with a standard deviation of 4.701 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, if you randomly select a landscaper in the US, what is the likelihood that his/her vitamin D level will be between 57.21 and 60.85 ng/mL?

2. Suppose at random 30% of school children develop nausea and vomiting following holiday parties and that you conduct a study to examine this phenomenon, with a sample size of n=43. What is the probability that 31 or more children become sick?
3. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 54.66 ng/mL with a standard deviation of 5.638 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, what is the lower value of the region for which approximately 68% of the data are located within the distribution? Recall the 68-95-99.9% observation discussed in the "Normal Distribution" section of the textbook that is helpful in answering the question.

4. Suppose at random 30% of school children develop nausea and vomiting following holiday parties and that you conduct a study to examine this phenomenon, with a sample size of n=28. What is the probability that between 13 or more and less than 24 children become sick?

5. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 35.18 ng/mL with a standard deviation of 5.744 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, if you randomly select a landscaper in the US, what is the probability that his/her vitamin D level will be between 21.89 and 49.69 ng/mL?

Answer 1

1. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 45.29 ng/mL with a standard deviation of 4.701 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, if you randomly select a landscaper in the US, what is the likelihood that his/her vitamin D level will be between 57.21 and 60.85 ng/mL?

Let X denotes the blood vitamin D levels of landscape gardeners in entire US population.

The likelihood that a randomly selected landscaper in the US has vitamin D level between 57.21 and 60.85 ng/mL is calculated using standard normal table and is given by:

Hence, the likelihood that a randomly selected landscaper in the US has vitamin D level between 57.21 and 60.85 ng/mL is 0.005.

2. Suppose at random 30% of school children develop nausea and vomiting following holiday parties and that you conduct a study to examine this phenomenon, with a sample size of n=43. What is the probability that 31 or more children become sick?

Given Information:

It is given that 30% of school children developed nausea and vomiting following holiday parties.

Let X denotes the number of school children who develop nausea and vomiting following holiday parties in a study conducted to examine this phenomenon for a sample size of 43.

The mean number of school children who develop nausea and vomiting following holiday parties is given by:

The standard deviation of number of school children who develop nausea and vomiting following holiday parties is given by:

and

The conditions for normal approximation to binomial are satisfied as np and nq are greater than equal to 5.

The probability that 31 or more children become sick can be approximated by normal distribution because the conditions for normal approximation to binomial is satisfied.

The probability that 31 or more children become sick is calculated by using a continuity correction factor of 0.5 and standard normal table and is given by:

Hence, the probability that 31 or more children become sick is 0.

1. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 54.66 ng/mL with a standard deviation of 5.638 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, what is the lower value of the region for which approximately 68% of the data are located within the distribution? Recall the 68-95-99.9% observation discussed in the "Normal Distribution" section of the textbook that is helpful in answering the question.

Let X denotes the blood vitamin D levels of landscape gardeners in entire US population.

The 68-95-99.9% rule of normal distribution states that:

The lower value of the region for which approximately 68% of the data are located within the distribution is calculated by using the lower limit of (1) and is given by:

Hence, the lower value of the region for which approximately 68% of the data are located within the distribution is 49.022.

1. Suppose at random 30% of school children develop nausea and vomiting following holiday parties and that you conduct a study to examine this phenomenon, with a sample size of n=28. What is the probability that between 13 or more and less than 24 children become sick?

It is given that 30% of school children developed nausea and vomiting following holiday parties.

Let X denotes the number of school children who develop nausea and vomiting following holiday parties in a study conducted to examine this phenomenon for a sample size of 28.

The mean number of school children who develop nausea and vomiting following holiday parties is given by:

The standard deviation of number of school children who develop nausea and vomiting following holiday parties is given by:

   and

The conditions for normal approximation to binomial are satisfied as np and nq are greater than equal to 5.

The probability that between 13 or more and less than 24 children beme sick is calculated by using standard normal table and is given by:

Hence, the probability that between 13 or more and less than 24 children become sick is 0.0287.

1. A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 35.18 ng/mL with a standard deviation of 5.744 ng/mL. Assuming the true distribution of blood vitamin D levels follows a Gaussian distribution, if you randomly select a landscaper in the US, what is the probability that his/her vitamin D level will be between 21.89 and 49.69 ng/mL?

Let X denotes the blood vitamin D levels of landscape gardeners in entire US population.

The probability that a randomly selected landscaper in the US has vitamin D level between 21.89 and 49.69 ng/mL is calculated using standard normal table and is given by:

Hence, the probability that a randomly selected landscaper in the US has vitamin D level between 21.89 and 49.69 ng/mL is 0.9839.