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 In a large population, 76% of the households own microwaves. A simple random sample of 100 households is to be contacted and the sample proportion computed.
In a large population, 76% of the households own microwaves. A simple random sample of 100 households is to be contacted and the sample proportion computed.
Learn the statistical parameters behind sampling distribution mean and standard deviation when examining household microwave ownership in a large population.
by Maivizhi A
Updated Mar 18, 2024
In a large population, 76% of the households own microwaves. A simple random sample of 100 households is to be contacted and the sample proportion computed. What is the mean and standard deviation of the sampling distribution of the sample proportions?
The mean (μ) of the sampling distribution of the sample proportions is 76%, and the standard deviation (σ_p) is approximately 0.0427.
Mean (μ): The mean of the sampling distribution of the sample proportions (phat) is equal to the population proportion (p).
Therefore, the mean (μ) is:
μ = p = 76%
Standard deviation (σ_p): The standard deviation of the sampling distribution of the sample proportions is calculated using the formula:
σ_p = sqrt(p * (1  p) / n)
where:
 p is the population proportion (0.76)
 n is the sample size (100)
Calculating the standard deviation:
σ_p = sqrt(0.76 * (1  0.76) / 100)
σ_p ≈ 0.0427
Therefore, the mean (μ) of the sampling distribution of the sample proportions is 76%, and the standard deviation (σ_p) is approximately 0.0427.
Mean and Standard Deviation
Mean and standard deviation are two important measures used in statistics to describe the distribution or spread of a set of data.

Mean: The mean, also known as the average, is calculated by adding up all the values in a dataset and then dividing by the total number of values. It represents the central tendency of the data.
Formula: Mean = (Sum of all values) / (Total number of values)

Standard Deviation: The standard deviation measures the dispersion or spread of the values in a dataset. It indicates how much the values differ from the mean. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.
Formula: Standard Deviation = Square root of [(Sum of squares of differences from the mean) / (Total number of values)]
The standard deviation is the square root of the variance, where the variance is the average of the squared differences from the Mean. It's often represented by the Greek letter sigma (σ) for population standard deviation and s for sample standard deviation.
In a large population, 76% of the households own microwaves. A simple random sample of 100 households is to be contacted and the sample proportion computed  FAQs
1. What is the population proportion of households owning microwaves?
The population proportion is 76%.
2. What does the mean (μ) represent in the sampling distribution of sample proportions?
The mean (μ) represents the expected proportion of households owning microwaves in the sample, which is also 76%.
3. How is the standard deviation of the sampling distribution calculated?
The standard deviation (σ_p) is calculated using the formula: sqrt(p * (1  p) / n), where p is the population proportion (0.76) and n is the sample size (100).
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