Question

Suppose x has a distribution with μ=65 and σ=10. Random samples of size 40 are drawn. Calculate the following probability. Round your answer to 4 decimal places.

P(xˉ>67)=
Suppose x has a distribution with μ=48 and σ=12. Random samples of size 73 are drawn. Calculate the following probability. Round your answer to 4 decimal places.
P(xˉ<47)=
Suppose x has a distribution with μ=58 and σ=8. Random samples of size 68 are drawn. Calculate the following probability. Round your answer to 4 decimal places.
P(57

Answer 1

1. For the first question,
`\( P(\bar{x} > 67) \),` calculate the z-score using the provided formula and find the probability using a standard normal distribution table.

2. For the second question,
`\( P(\bar{x} < 47) \)` calculate the z-score using the given formula and determine the probability using a standard normal distribution table.

Step-by-step explanation:

To calculate these probabilities, you can use the z-score formula and standard normal distribution tables.

The formula for the z-score is given by:

`\[ z = \frac{{\bar{x} - \mu}}{{(\sigma)/(√(n))}} \]`

where:

-
`\( \bar{x} \)`is the sample mean,

-
`\( \mu \)` is the population mean,

-
`\( \sigma \)` is the population standard deviation,

-
`\( n \)` is the sample size.

For the first question:

`\[ P(\bar{x} > 67) \]`

`\[ z = \frac{{67 - 65}}{{(10)/(√(40))}} \]`

Calculate the z-score and then find the corresponding probability using a standard normal distribution table.

For the second question:

`\[ P(\bar{x} < 47) \]`

`\[ z = \frac{{47 - 48}}{{(12)/(√(73))}} \]`

Calculate the z-score and find the corresponding probability.

For the third question:

`\[ P(\bar{x} > \text{some value}) \]`

`\[ z = \frac{{\text{some value} - 58}}{{(8)/(√(68))}} \]`

Calculate the z-score and find the corresponding probability.

Please note that you may need to use a standard normal distribution table or a calculator with statistical functions to find the probabilities associated with the calculated z-scores.