In a survey, $$ 40 \% $$ of people prefer chocolate ice cream. If 200 people are surveyed. robability that at least 90 people prefer chocolate ice cream.

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UpStudy AI · Accepted Answer

To find the probability that at least 90 people prefer chocolate ice cream out of 200 surveyed, we can model this situation using a binomial distribution.

### Known Conditions:
- The probability of success (preferring chocolate ice cream) $$ p = 0.4 $$
- The number of trials (people surveyed) $$ n = 200 $$
- We want to find $$ P(X \geq 90) $$, where $$ X $$ is the number of people who prefer chocolate ice cream.

### Steps to Solve:
1. **Calculate the mean and standard deviation of the binomial distribution:**
   - Mean $$ \mu = n \cdot p $$
   - Standard deviation $$ \sigma = \sqrt{n \cdot p \cdot (1 - p)} $$

2. **Use the normal approximation to the binomial distribution:**
   Since $$ n $$ is large, we can use the normal approximation:
   - Convert $$ X $$ to a standard normal variable $$ Z $$:
   $$
   Z = \frac{X - \mu}{\sigma}
   $$

3. **Calculate $$ P(X \geq 90) $$ using the normal distribution:**
   - This is equivalent to finding $$ P(Z \geq z) $$.

Let's perform the calculations step by step.

### Step 1: Calculate Mean and Standard Deviation
- Mean $$ \mu = 200 \cdot 0.4 $$
- Standard deviation $$ \sigma = \sqrt{200 \cdot 0.4 \cdot (1 - 0.4)} $$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$200\times 0.4$$
- step1: Multiply the numbers:
  $$80$$
Calculate or simplify the expression $$ \sqrt(200 * 0.4 * (1 - 0.4)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{200\times 0.4\left(1-0.4\right)}$$
- step1: Subtract the numbers:
  $$\sqrt{200\times 0.4\times 0.6}$$
- step2: Multiply the terms:
  $$\sqrt{48}$$
- step3: Write as a product:
  $$\sqrt{16\times 3}$$
- step4: Write in exponential form:
  $$\sqrt{4^{2}\times 3}$$
- step5: Use the properties of radicals:
  $$\sqrt{4^{2}}\times \sqrt{3}$$
- step6: Simplify the root:
  $$4\sqrt{3}$$
### Step 1 Results:
- Mean $$ \mu = 80 $$
- Standard deviation $$ \sigma \approx 6.93 $$

### Step 2: Convert to Standard Normal Variable
Now, we need to find $$ z $$ for $$ X = 90 $$:
$$
z = \frac{90 - 80}{6.93}
$$

Let's calculate $$ z $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(90-80\right)}{6.93}$$
- step1: Remove the parentheses:
  $$\frac{90-80}{6.93}$$
- step2: Subtract the numbers:
  $$\frac{10}{6.93}$$
- step3: Convert the expressions:
  $$\frac{10}{\frac{693}{100}}$$
- step4: Multiply by the reciprocal:
  $$10\times \frac{100}{693}$$
- step5: Multiply:
  $$\frac{10\times 100}{693}$$
- step6: Multiply:
  $$\frac{1000}{693}$$
### Step 2 Result:
- The calculated $$ z $$ value is approximately $$ 1.44 $$.

### Step 3: Calculate $$ P(Z \geq 1.44) $$
To find $$ P(X \geq 90) $$, we need to find $$ P(Z \geq 1.44) $$. This can be found using the standard normal distribution table or a calculator.

Let's calculate $$ P(Z \geq 1.44) $$. This is equivalent to $$ 1 - P(Z < 1.44) $$.

Now, I will calculate $$ P(Z < 1.44) $$.
It seems I cannot directly calculate the cumulative probability for the standard normal distribution. However, I can provide the approximate value based on standard normal distribution tables.

### Approximate Value:
From standard normal distribution tables, we find:
- $$ P(Z < 1.44) \approx 0.9251 $$

Thus, we can calculate:
$$
P(Z \geq 1.44) = 1 - P(Z < 1.44) \approx 1 - 0.9251 = 0.0749
$$

### Final Result:
The probability that at least 90 people prefer chocolate ice cream is approximately $$ 0.0749 $$ or $$ 7.49\% $$.
The probability that at least 90 people prefer chocolate ice cream is approximately 7.49%.

In a survey, of people prefer chocolate ice cream. If 200 people are surveyed.
robability that at least 90 people prefer chocolate ice cream.

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In a survey, of people prefer chocolate ice cream. If 200 people are surveyed. robability that at least 90 people prefer chocolate ice cream.