Question

The number of chocolate chips in a single chocolate chip cookie is approximately normally distributed with a mean of

12.5 chocolate chips and a standard deviation of 2.1 chocolate chips. What is the probability that a randomly selected
chocolate chip cookie contains fewer than 10 chocolate chips?
Your answer should be a decimal value rounded to four decimal places.
The probability is

Answer 1

To solve this problem, we need to standardize the value 10 using the formula:

z = (x - μ) / σ

where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

z = (10 - 12.5) / 2.1 = -1.19

Using a standard normal table or calculator, we can find that the probability of a standard normal variable being less than -1.19 is 0.1179.

Therefore, the probability that a randomly selected chocolate chip cookie contains fewer than 10 chocolate chips is 0.1179 or 11.79%.

hope it helps you....

Answer 2

0.1170 (using z-tables)

0.1169 (using a calculator)

Explanation:

The number of chocolate chips in a single chocolate chip cookie is approximately normally distributed with a mean of 12.5 chocolate chips and a standard deviation of 2.1 chocolate chips.

Therefore:

`\boxed{X \sim\text{N} \left(12.5,2.1^2\right)}`

where X is the number of chocolate chips in a single chocolate chip cookie.

To find the probability that a randomly selected chocolate chip cookie contains fewer than 10 chocolate chips, we need to find P(X < 10).

Converting to the Z distribution:

`\boxed{\textsf{If }\: X \sim\textsf{N}(\mu,\sigma^2)\:\textsf{ then }\: (X-\mu)/(\sigma)=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)}`

Transform X to Z:

`\begin{aligned}\text{P}(X < 10) &= \text{P}\left(Z < (10-12.5)/(2.1)\right)\\\\&=\text{P}(Z < -1.19047619...)\end{aligned}`

According to the z-tables, when Z = -1.19, p = 0.11702.

Therefore:

`\text{P}(X < 10) =0.1170\;\sf(4\;d.p.)`

Therefore, probability that a randomly selected chocolate chip cookie contains fewer than 10 chocolate chips is 0.1170.

`\hrulefill`

Additional Information

If we use a calculator to calculate the "normal cumulative distribution function (cdf)", we get a slightly different (more accurate) result than when using z-tables.

Input the following parameters into the "normal cumulative distribution function (cdf)" of a calculator:

• Upper bound: x = 10
• Lower bound: x = -100
• μ = 12.5
• σ =2.1

Therefore:

`\text{P}(X < 10) =0.116929641...`

`\text{P}(X < 10) =0.1169\; \sf (4\;d.p.)`

Therefore, probability that a randomly selected chocolate chip cookie contains fewer than 10 chocolate chips is 0.1169.