Math / Data & Statistics

QuestionA weight-loss clinic guarantees that its new customers will lose at least 5 lb by the end of their first month of participation or their money will be refunded. If the loss of weight of customers at the end of their first month is normally distributed, with a mean of 6.2 lb and a standard deviation of 0.81 lb , find the percent of customers who will be able to claim a refund. Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. $\square$ \% of customers will be able to claim a refund. (Round to the nearest tenth as needed.)

Studdy Solution

STEP 1

What is this asking? What percentage of people lost less than 5 lbs? Watch out! Don't forget to convert to a z-score before using the z-table!

STEP 2

1. Calculate the z-score.
2. Find the probability using the z-table.
3. Convert to percentage.

STEP 3

Alright, so we're dealing with a normal distribution here, which means we can use our handy-dandy z-scores!
The z-score tells us how many standard deviations a particular value is away from the mean.
A negative z-score means the value is below the mean, and a positive z-score means the value is above the mean.

STEP 4

The formula for the z-score is: $z = \frac{x - \mu}{\sigma}$ Where $x$ is our value of interest (in this case, 5 lbs), $\mu$ is the mean (6.2 lbs), and $\sigma$ is the standard deviation (0.81 lbs).

STEP 5

Let's plug in our values: $z = \frac{5 - 6.2}{0.81}$ $z = \frac{-1.2}{0.81}$ $z \approx -1.48$ So, our z-score is approximately -1.48.
This means 5 lbs is 1.48 standard deviations below the mean.

STEP 6

Now, we grab our z-table!
We're looking for the probability of a z-score being less than -1.48, which represents the percentage of customers who lost less than 5 lbs.

STEP 7

Looking up -1.48 in the z-table, we find the probability to be approximately 0.0694.
This means there's about a 6.94% chance that a randomly selected customer lost less than 5 lbs.

STEP 8

Finally, we convert our probability to a percentage by multiplying by 100: $0.0694 \cdot 100 = 6.94\%$ Rounding to the nearest tenth gives us 6.9%.

STEP 9

Approximately 6.9% of customers will be able to claim a refund.

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Studdy Solution

STEP 1

What is this asking? What percentage of people lost less than 5 lbs? Watch out! Don't forget to convert to a z-score before using the z-table!

STEP 2

1. Calculate the z-score.
2. Find the probability using the z-table.
3. Convert to percentage.

STEP 3

STEP 4

The formula for the z-score is: $z = \frac{x - \mu}{\sigma}$ Where $x$ is our value of interest (in this case, 5 lbs), $\mu$ is the mean (6.2 lbs), and $\sigma$ is the standard deviation (0.81 lbs).

STEP 5

Let's plug in our values: $z = \frac{5 - 6.2}{0.81}$ $z = \frac{-1.2}{0.81}$ $z \approx -1.48$ So, our z-score is approximately -1.48.
This means 5 lbs is 1.48 standard deviations below the mean.

STEP 6

Now, we grab our z-table!
We're looking for the probability of a z-score being less than -1.48, which represents the percentage of customers who lost less than 5 lbs.

STEP 7

Looking up -1.48 in the z-table, we find the probability to be approximately 0.0694.
This means there's about a 6.94% chance that a randomly selected customer lost less than 5 lbs.

STEP 8

Finally, we convert our probability to a percentage by multiplying by 100: $0.0694 \cdot 100 = 6.94\%$ Rounding to the nearest tenth gives us 6.9%.

STEP 9

Approximately 6.9% of customers will be able to claim a refund.

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Studdy Solution

STEP 1

What is this asking? What percentage of people lost *less* than 5 lbs? Watch out! Don't forget to convert to a z-score *before* using the z-table!

STEP 2

1. Calculate the z-score.2. Find the probability using the z-table.3. Convert to percentage.

STEP 3

STEP 4

The formula for the z-score is: z=x−μσ z = \frac{x - \mu}{\sigma} z=σx−μ​ Where xxx is our **value of interest** (in this case, 5 lbs), μ\muμ is the **mean** (6.2 lbs), and σ\sigmaσ is the **standard deviation** (0.81 lbs).

STEP 5

Let's plug in our values: z=5−6.20.81 z = \frac{5 - 6.2}{0.81} z=0.815−6.2​ z=−1.20.81 z = \frac{-1.2}{0.81} z=0.81−1.2​z≈−1.48 z \approx -1.48 z≈−1.48So, our **z-score** is approximately **-1.48**.This means 5 lbs is 1.48 standard deviations *below* the mean.

STEP 6

Now, we grab our z-table!We're looking for the probability of a z-score being *less than* -1.48, which represents the percentage of customers who lost *less than* 5 lbs.

STEP 7

Looking up -1.48 in the z-table, we find the probability to be approximately **0.0694**.This means there's about a 6.94% chance that a randomly selected customer lost less than 5 lbs.

STEP 8

Finally, we convert our probability to a percentage by multiplying by 100: 0.0694⋅100=6.94% 0.0694 \cdot 100 = 6.94\% 0.0694⋅100=6.94% Rounding to the nearest tenth gives us **6.9%**.

STEP 9

Approximately **6.9%** of customers will be able to claim a refund.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? What percentage of people lost less than 5 lbs? Watch out! Don't forget to convert to a z-score before using the z-table!

1. Calculate the z-score.
2. Find the probability using the z-table.
3. Convert to percentage.

The formula for the z-score is: $z = \frac{x - \mu}{\sigma}$ Where $x$ is our value of interest (in this case, 5 lbs), $\mu$ is the mean (6.2 lbs), and $\sigma$ is the standard deviation (0.81 lbs).

Let's plug in our values: $z = \frac{5 - 6.2}{0.81}$ $z = \frac{-1.2}{0.81}$ $z \approx -1.48$ So, our z-score is approximately -1.48.
This means 5 lbs is 1.48 standard deviations below the mean.

Now, we grab our z-table!
We're looking for the probability of a z-score being less than -1.48, which represents the percentage of customers who lost less than 5 lbs.

Looking up -1.48 in the z-table, we find the probability to be approximately 0.0694.
This means there's about a 6.94% chance that a randomly selected customer lost less than 5 lbs.

Finally, we convert our probability to a percentage by multiplying by 100: $0.0694 \cdot 100 = 6.94\%$ Rounding to the nearest tenth gives us 6.9%.

Approximately 6.9% of customers will be able to claim a refund.