Question

You are interested in the average monthly cell phone bill for US households (Assume cell phone bills have a normal distribution). You select a random sample of 61 households and find the average monthly cell phone bill is $174.95. You calculate the sample standard deviation is $41.40.

Estimate the average monthly cell phone bill for US households based on your data using formulas. Show any formulas needed, show all work and use proper units. Remember that ALL estimates should include an appropriate interpretation! Use 95% confidence level and round the final answer to two decimal places.

Answer 1

We are 95% confident that the average monthly cell phone bill for US households is between $162.61 and $187.29.

We can estimate the population mean
`\mu` with a confidence interval. The formula for the confidence interval is:

`\bar{x} \pm t_(\alpha / 2, d f) \cdot (s)/(√(n))`

where:

-
`$\bar{x}$` is the sample mean

-
`$t_(\alpha / 2, d _f)$` is the t-value at the
`\alpha / 2` significance level with
`d _f` degrees of freedom

- s is the sample standard deviation

- n is the sample size

Assumptions:

We assume that the population of monthly cell phone bills is normally distributed.

We are using a 95% confidence level, which means α=0.05.

The sample size is large enough (n > 30) to use the t-distribution.

Python

# Sample size

n = 61

# Sample mean

x_bar = 174.95

# Sample standard deviation

s = 41.40

# Confidence level

confidence_level = 0.95

# Degrees of freedom

df = n - 1

# t-value

t_value = np.abs(np.quantile(np.abs(np.random.standard_t(df, size=10000)), 1 - (1 - confidence_level) / 2))

# Standard error

se = s / np.sqrt(n)

# Confidence interval

margin_of_error = t_value * se

lower_bound = x_bar - margin_of_error

upper_bound = x_bar + margin_of_error

print(f"We are 95% confident that the average monthly cell phone bill for US households is between ${lower_bound:.2f} and ${upper_bound:.2f}.")

Interpretation:

We are 95% confident that the true average monthly cell phone bill for US households is between $162.61 and $187.29. In other words, if we were to take many random samples of 61 households each, and calculate the average monthly cell phone bill for each sample, 95% of those samples would have an average between $162.61 and $187.29.