$1.$ The design of a package or bottle can have an important effect on a company$$s bottom line. In this case a brand group wishes to research consumer reaction to a new bottle design for a popular soft drink. Because it is impossible to show the new bottle design to $$all consumers$,$ the brand group will use mall intercept method to select a sample of $60$ consumers. For each consumer interviewed, a bottle image composite score will be found by adding the consumer$$s numerical responses to the five questions. It follows that the minimum possible bottle image composite score is $5($resulting from a response of $1$ on all five questions$)$ and the maximum bottle image composite score is $35($resulting from a response of $7$ on all five questions$).$ The scores of $60$ consumers is given below: $(20$ marks$)$

$34333329263328253233$

$32252733222732333229$

$24302034313230353331$

$32283031313329273431$

$31283331322826293234$

$32303432303032312933$

a$.$ Develop a $95\%$ confidence interval for the population composite score.

b$.$ Furthermore, experience has shown that the smallest acceptable bottle image score for a successful bottle design is $29.$ Test the hypothesis at $5\%$ significance level that the new design will be accepted by the consumers.