A random sample of size n=169 yielded p̂=.65. a. Is the sample size large enough to use the meth

step 1 For this problem, we are told that a random sample of size n equals 169 yielded p hat equals 0 .

A random sample of size n=169 yielded p̂=.65. a. Is the...

A random sample of size $n=169$ yielded $\hat{p}=.65$. a. Is the sample size large enough to use the methods of this section to construct a confidence interval for $p ?$ Explain. b. Construct a $90 \%$ confidence interval for $p$. c. What assumption is necessary to ensure the validity of this confidence interval?

A random sample of size $n=169$ yielded $\hat{p}=.65$.
a. Is the sample size large enough to use the methods of this section to construct a confidence interval for $p ?$ Explain.
b. Construct a $90 \%$ confidence interval for $p$.
c. What assumption is necessary to ensure the validity of this confidence interval?

Key Concepts

Sample Size Considerations

This concept involves checking whether the sample is large enough to reliably use normal approximations. For proportions, the rule of thumb is that both np? and n(1-p?) should be at least 10. This condition ensures that the sampling distribution of the proportion is approximately normal, which is crucial for valid inferential methods.

Normal Approximation to the Binomial Distribution

When the sample size is large, the binomial distribution of a proportion can be approximated by a normal distribution. This is a direct consequence of the Central Limit Theorem and allows one to use z-scores for constructing confidence intervals around the sample proportion.

Confidence Interval Construction for a Proportion

This key concept refers to the process of estimating the true population proportion by constructing an interval around the sample proportion. The interval is generally calculated as p? ± (z-value × standard error), where the standard error is derived from the sample proportion and the sample size.

Underlying Assumptions

The validity of the confidence interval depends on a set of assumptions including that the sample is randomly selected, the data points are independent, and that the successes and failures meet minimal count conditions for the normal approximation to hold. These assumptions ensure the interval accurately reflects the population parameters.

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