California has a population of approximately 528,120 U.S. Indians. In a 2012 survey of Asian Americans, Pew Research Center found that \( 70 \% \) of U.S. Indians celebrate Diwali from a sample size of 580 . (a) What is the estimated mean of the sampling distribution of sample proportions of this same sample size? \( p=528120 \) \( \hat{p}=0.70 \) \( \mu_{\hat{p}}=0.70 \) \( p=0.70 \) \( \sqrt{\frac{0.70(1-0.70)}{580}} \) What is the standard error for the sampling distribution of sample proportions of this same sample size? (b) \( \sqrt{\frac{528120(1-528120)}{580}} \) \( \sqrt{\frac{580(1-580)}{528120}} \)

The estimated mean of the sampling distribution of sample proportions is calculated using the sample proportion \(\hat{p}\). Given that \( \hat{p} = 0.70 \), the mean of the sampling distribution, \( \mu_{\hat{p}} \), is simply \( 0.70 \). This means that if we repeatedly took samples of size 580, we'd expect about 70% of the sample to celebrate Diwali on average. To find the standard error (SE) for the sampling distribution of sample proportions, the correct formula is given by: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.70(1-0.70)}{580}} = \sqrt{\frac{0.70 \cdot 0.30}{580}} \approx 0.039. \] This tells us how much the sample proportion is likely to vary from the true population proportion of those celebrating Diwali. When diving into the historical significance of Diwali, it’s fascinating to note that this festival of lights has been celebrated for thousands of years, originating from Hindu tradition but now embraced by various cultures. Over time, it has become a symbol of joy and the triumph of light over darkness, drawing millions together in grand celebrations each year. As for real-world applications, the understanding of sampling distributions and standard errors is crucial in various fields, especially in business and social sciences. Analysts use these concepts to interpret survey results, gauge public opinion, or develop marketing strategies. A well-calculated standard error helps in making informed decisions based on sample data without jumping to conclusions that could lead to costly mistakes.