A study is performed in El Paso to determine whether the average weekly grocery bill per five-person family in the town is significantly different from the national average of $130. A random sample of 56 five-person families in El Paso showed a mean of $134.47. There is considerable variability in the amount and historically the standard deviation has been $11.19.

(a)Construct a 95% confidence interval for the mean.

(b) Does this interval indicate that the mean for this shop differs from the national average? Explain.

(c)What is the length of the interval? If we decreased the confidence level to 90%, what would happen to the length of the interval?

(d) What would be the necessary change to the sample size if we want to cut the length in half?

(e) If the standard deviation of $11.19 was from the sample and not the historical standard deviation, how would the interval change? Explain.