A nationwide study indicated that 27% of adults said that chocolate was their favorite ice cream flavor. A simple random sample of 150 adults is obtained.
A.) Describe the sampling distribtuion of p, the sample proportion of adults whose favorite ice cream flavor is chocolate. State the type of distribution, find its mean and the standard deviation.
B.) In a random sample of 150 adults, what is the probability tha at most 22% will say that their favorite ice cream flavor is chocolate?
C.) In a random sample of 150 adults, what is the probability that at most 22% and 30% will say that their favorite ice cream flavor is chocolate?
D.) Would it be unusual if a random sample of 150 adults reulted in 27 people that said that their favorite ice cream flavor was chocolate? Explain why yes or why no.
In a box of 100 cookies, 36 contain chocolate and 12 contain nuts. Of those, 8 cookies contain
A.) The sampling distribution of p, the sample proportion of adults whose favorite ice cream flavor is chocolate, can be approximated by a normal distribution. This is because the sample size (150 adults) is large enough for the Central Limit Theorem to apply, which states that the distribution of sample means, or proportions in this case, tends to be approximately normal regardless of the shape of the population distribution.
The mean of the sampling distribution of p is equal to the population proportion, which is 0.27 (27%). The standard deviation can be calculated using the formula:
Standard Deviation (σ) = sqrt((p * (1 - p)) / n),
where p is the population proportion (0.27) and n is the sample size (150). Plugging the values into the formula, we get:
σ = sqrt((0.27 * (1 - 0.27)) / 150) ≈ 0.0329 (rounded to 4 decimal places).
So, the mean of the sampling distribution is 0.27 and the standard deviation is approximately 0.0329.
B.) To find the probability that at most 22% will say that their favorite ice cream flavor is chocolate, we need to calculate the probability that the sample proportion (p) is less than or equal to 0.22.
First, we need to standardize the value of 0.22 by subtracting the mean and dividing by the standard deviation. Using the formula for z-score:
z = (x - μ) / σ,
where x is the value we want to standardize (0.22), μ is the mean of the sampling distribution (0.27), and σ is the standard deviation of the sampling distribution (0.0329), we get:
z = (0.22 - 0.27) / 0.0329 ≈ -1.52 (rounded to 2 decimal places).
Next, we need to find the probability corresponding to this standardized value (z) from the standard normal distribution table or by using a calculator. Looking up the value of -1.52 in the table, we find that the probability is approximately 0.0630.
Therefore, the probability that at most 22% will say that their favorite ice cream flavor is chocolate is approximately 0.0630, or 6.3%.
C.) To find the probability that at most 22% and 30% will say that their favorite ice cream flavor is chocolate, we need to calculate the probability that the sample proportion (p) is less than or equal to 0.22 and less than or equal to 0.30.
By following the same steps as in part B, you can find the probability for each individual value (0.22 and 0.30) and then calculate the probability of both events occurring together by multiplying the individual probabilities.
D.) No, it would not be unusual if a random sample of 150 adults resulted in 27 people saying that their favorite ice cream flavor was chocolate. This is because the expected value (mean) of the sample proportion is 0.27, which means that, on average, we would expect 27% of adults to prefer chocolate. Therefore, a result of exactly 27 people out of 150 saying chocolate is their favorite ice cream flavor would be in line with the expected value.
However, keep in mind that this assumes the sample was randomly selected and represents a diverse population of adults. If there are any biases or limitations in the sampling method, then the result may not be truly representative of the population.